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2401.09350 | 44 | Vector representations of objects have long been an integral part of the machine learning literature. Indeed, a classifier, a regression model, or a rank- ing function learns patterns from, and acts on, vector representations of data. In the past, this vector representation of an object was nothing more than a collection of its features. Every feature described some facet of the object (for example, the color intensity of a pixel in a photograph) as a continuous or discrete value. The idea was that, while individual features describe only a small part of the object, together they provide sufficiently powerful statistics about the object and its properties for the machine learnt model to act on. The features that led to the vector representation of an object were gen- erally hand-crafted functions. To make sense of that, let us consider a text document in English. Strip the document of grammar and word order, and
3
4
1 Vector Retrieval
brown Pox brown Fox jumps | ota o fr =O ee = os 2D The quick brown Fox â>(0_0 1% = 0 jumps over the lazy dog... Zs games over une 1027 OG") The five boxing wizards op jume auickly... (00 7% 0 = 0 = Wo) | 2401.09350#44 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 44,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Vector representations of objects have long been an integral part of the machine learning literature. Indeed, a classifier, a regression model, or a rank- ing function learns patterns from, and acts on, vector representations of data. In the past, this vector representation of an object was nothing more than a collection of its features. Every feature described some facet of the object (for example, the color intensity of a pixel in a photograph) as a continuous or discrete value. The idea was that, while individual features describe only a small part of the object, together they provide sufficiently powerful statistics about the object and its properties for the machine learnt model to act on. The features that led to the vector representation of an object were gen- erally hand-crafted functions. To make sense of that, let us consider a text document in English. Strip the document of grammar and word order, and\n3\n4\n1 Vector Retrieval\nbrown Pox brown Fox jumps | ota o fr =O ee = os 2D The quick brown Fox â>(0_0 1% = 0 jumps over the lazy dog... Zs games over une 1027 OG\") The five boxing wizards op jume auickly... (00 7% 0 = 0 = Wo)",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 45 | Fig. 1.1: Vector representation of a piece of text by adopting a âbag of wordsâ view: A text document, when stripped of grammar and word order, can be thought of as a vector, where each coordinate represents a term in our vo- cabulary and its value records the frequency of that term in the document or some function of it. The resulting vectors are typically sparse; that is, they have very few non-zero coordinates.
we end up with a set of words, more commonly known as a âbag of words.â This set can be summarized as a histogram.
If we designated every term in the English vocabulary to be a dimension in a (naturally) high-dimensional space, then the histogram representation of the document can be encoded as a vector. The resulting vector has relatively few non-zero coordinates, and each non-zero coordinate records the frequency of a term present in the document. This is illustrated in Figure 1.1 for a toy example. More generally, non-zero values may be a function of a termâs frequency in the document and its propensity in a collectionâthat is, the likelihood of encountering the term [Salton and Buckley, 1988]. | 2401.09350#45 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 45,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 1.1: Vector representation of a piece of text by adopting a âbag of wordsâ view: A text document, when stripped of grammar and word order, can be thought of as a vector, where each coordinate represents a term in our vo- cabulary and its value records the frequency of that term in the document or some function of it. The resulting vectors are typically sparse; that is, they have very few non-zero coordinates.\nwe end up with a set of words, more commonly known as a âbag of words.â This set can be summarized as a histogram.\nIf we designated every term in the English vocabulary to be a dimension in a (naturally) high-dimensional space, then the histogram representation of the document can be encoded as a vector. The resulting vector has relatively few non-zero coordinates, and each non-zero coordinate records the frequency of a term present in the document. This is illustrated in Figure 1.1 for a toy example. More generally, non-zero values may be a function of a termâs frequency in the document and its propensity in a collectionâthat is, the likelihood of encountering the term [Salton and Buckley, 1988].",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 46 | The advent of deep learning and, in particular, Transformer-based mod- els [Vaswani et al., 2017] brought about vector representations that are be- yond the elementary formation above. The resulting representation is often, as a single entity, referred to as an embedding, instead of a âfeature vector,â though the underlying concept remains unchanged: an object is encoded as a real d-dimensional vector, a point in Rd.
Let us go back to the example from earlier to see how the embedding of a text document could be different from its representation as a frequency-based feature vector. Let us maintain the one-to-one mapping between coordinates and terms in the English vocabulary. Remember that in the âlexicalâ rep- resentation from earlier, if a coordinate was non-zero, that implied that the corresponding term was present in the document and its value indicated its frequency-based feature. Here we instead learn to turn coordinates on or off and, when we turn a coordinate on, we want its value to predict the significance of the corresponding term based on semantics and contextual in- formation. For example, the (absent) synonyms of a (present) term may get a non-zero value, and terms that offer little discriminative power in the given context become 0 or close to it. This basic idea has been explored extensively
1.2 Vectors as Units of Retrieval | 2401.09350#46 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 46,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The advent of deep learning and, in particular, Transformer-based mod- els [Vaswani et al., 2017] brought about vector representations that are be- yond the elementary formation above. The resulting representation is often, as a single entity, referred to as an embedding, instead of a âfeature vector,â though the underlying concept remains unchanged: an object is encoded as a real d-dimensional vector, a point in Rd.\nLet us go back to the example from earlier to see how the embedding of a text document could be different from its representation as a frequency-based feature vector. Let us maintain the one-to-one mapping between coordinates and terms in the English vocabulary. Remember that in the âlexicalâ rep- resentation from earlier, if a coordinate was non-zero, that implied that the corresponding term was present in the document and its value indicated its frequency-based feature. Here we instead learn to turn coordinates on or off and, when we turn a coordinate on, we want its value to predict the significance of the corresponding term based on semantics and contextual in- formation. For example, the (absent) synonyms of a (present) term may get a non-zero value, and terms that offer little discriminative power in the given context become 0 or close to it. This basic idea has been explored extensively\n1.2 Vectors as Units of Retrieval",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 47 | 1.2 Vectors as Units of Retrieval
by many recent models of text [Bai et al., 2020, Formal et al., 2021, 2022, Zhuang and Zuccon, 2022, Dai and Callan, 2020, Gao et al., 2021, Mallia et al., 2021, Zamani et al., 2018, Lin and Ma, 2021] and has been shown to produce effective representations.
Vector representations of text need not be sparse. While sparse vectors with dimensions that are grounded in the vocabulary are inherently in- terpretable, text documents can also be represented with lower-dimensional dense vectors (where every coordinate is almost surely non-zero). This is, in fact, the most dominant form of vector representation of text documents in the literature [Lin et al., 2021, Karpukhin et al., 2020, Xiong et al., 2021, Reimers and Gurevych, 2019, Santhanam et al., 2022, Khattab and Zaharia, 2020]. Researchers have also explored hybrid representations of text where vectors have a dense subspace and an orthogonal sparse subspace [Chen et al., 2022, Bruch et al., 2023, Wang et al., 2021, Kuzi et al., 2020, Karpukhin et al., 2020, Ma et al., 2021, 2020, Wu et al., 2019]. | 2401.09350#47 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 47,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "1.2 Vectors as Units of Retrieval\nby many recent models of text [Bai et al., 2020, Formal et al., 2021, 2022, Zhuang and Zuccon, 2022, Dai and Callan, 2020, Gao et al., 2021, Mallia et al., 2021, Zamani et al., 2018, Lin and Ma, 2021] and has been shown to produce effective representations.\nVector representations of text need not be sparse. While sparse vectors with dimensions that are grounded in the vocabulary are inherently in- terpretable, text documents can also be represented with lower-dimensional dense vectors (where every coordinate is almost surely non-zero). This is, in fact, the most dominant form of vector representation of text documents in the literature [Lin et al., 2021, Karpukhin et al., 2020, Xiong et al., 2021, Reimers and Gurevych, 2019, Santhanam et al., 2022, Khattab and Zaharia, 2020]. Researchers have also explored hybrid representations of text where vectors have a dense subspace and an orthogonal sparse subspace [Chen et al., 2022, Bruch et al., 2023, Wang et al., 2021, Kuzi et al., 2020, Karpukhin et al., 2020, Ma et al., 2021, 2020, Wu et al., 2019].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 48 | Unsurprisingly, the same embedding paradigm can be extended to other data modalities beyond text: Using deep learning models, one may embed images, videos, and audio recordings into vectors. In fact, it is even possible to project different data modalities (e.g., images and text) together into the same vector space and preserve some property of interest [Zhang et al., 2020, Guo et al., 2019].
It appears, then, that vectors are everywhere. Whether they are the result of hand-crafted functions that capture features of the data or are the output of learnt models; whether they are dense, sparse, or both, they are effective representations of data of any modality.
But what precisely is the point of turning every piece of data into a vector? One answer to that question takes us to the fascinating world of retrieval.
# 1.2 Vectors as Units of Retrieval
It would make for a vapid exercise if all we had were vector representations of data without any structure governing a collection of them. To give a collection of points some structure, we must first ask ourselves what goal we are trying to achieve by turning objects into vectors. It turns out, we often intend for the vector representation of two similar objects to be âcloseâ to each other according to some well-defined distance function. | 2401.09350#48 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 48,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Unsurprisingly, the same embedding paradigm can be extended to other data modalities beyond text: Using deep learning models, one may embed images, videos, and audio recordings into vectors. In fact, it is even possible to project different data modalities (e.g., images and text) together into the same vector space and preserve some property of interest [Zhang et al., 2020, Guo et al., 2019].\nIt appears, then, that vectors are everywhere. Whether they are the result of hand-crafted functions that capture features of the data or are the output of learnt models; whether they are dense, sparse, or both, they are effective representations of data of any modality.\nBut what precisely is the point of turning every piece of data into a vector? One answer to that question takes us to the fascinating world of retrieval.\n# 1.2 Vectors as Units of Retrieval\nIt would make for a vapid exercise if all we had were vector representations of data without any structure governing a collection of them. To give a collection of points some structure, we must first ask ourselves what goal we are trying to achieve by turning objects into vectors. It turns out, we often intend for the vector representation of two similar objects to be âcloseâ to each other according to some well-defined distance function.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 49 | That is the structure we desire: Similarity in the vector space must imply similarity between objects. So, as we engineer features to be extracted from an object, or design a protocol to learn a model to produce embeddings of data, we must choose the dimensionality d of the target space (a subset of Rd)
5
6
1 Vector Retrieval
along with a distance function δ(·, ·). Together, these define an inner product or metric space.
Consider again the lexical representation of a text document where d is the size of the English vocabulary. Let δ be the distance variant of the Jaccard index, δ(u, v) = âJ(u, v) â â|nz (u) â© nz (v)|/|nz (u) ⪠nz (v)|, where nz (u) = {i | ui ̸= 0} with ui denoting the i-th coordinate of vector u.
In the resulting space, if vectors u and v have a smaller distance than vec- tors u and w, then we can clearly conclude that the document represented by u is lexically more similar to the one represented by v than it is to the document w represents. That is because the distance (or, in this case, simi- larity) function reflects the amount of overlap between the terms present in one document with another. | 2401.09350#49 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 49,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "That is the structure we desire: Similarity in the vector space must imply similarity between objects. So, as we engineer features to be extracted from an object, or design a protocol to learn a model to produce embeddings of data, we must choose the dimensionality d of the target space (a subset of Rd)\n5\n6\n1 Vector Retrieval\nalong with a distance function δ(·, ·). Together, these define an inner product or metric space.\nConsider again the lexical representation of a text document where d is the size of the English vocabulary. Let δ be the distance variant of the Jaccard index, δ(u, v) = âJ(u, v) â â|nz (u) â© nz (v)|/|nz (u) ⪠nz (v)|, where nz (u) = {i | ui ̸= 0} with ui denoting the i-th coordinate of vector u.\nIn the resulting space, if vectors u and v have a smaller distance than vec- tors u and w, then we can clearly conclude that the document represented by u is lexically more similar to the one represented by v than it is to the document w represents. That is because the distance (or, in this case, simi- larity) function reflects the amount of overlap between the terms present in one document with another.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 50 | We should be able to make similar arguments given a semantic embedding of text documents. Again consider the sparse embeddings with d being the size of the vocabulary, and more concretely, take SPLADE [Formal et al., 2021] as a concrete example. This model produces real-valued sparse vectors in an inner product space. In other words, the objective of its learning procedure is to maximize the inner product between similar vectors, where the inner product between two vectors u and v is denoted by (u,v) and is computed using >>, uri.
In the resulting space, if u, v, and w are generated by Splade with the property that â¨u, vâ© > â¨u, wâ©, then we can conclude that, according to Splade, documents represented by u and v are semantically more similar to each other than u is to w. There are numerous other examples of models that optimize for the angular distance or Euclidean distance (L2) between vectors to preserve (semantic) similarity.
What can we do with a well-characterized collection of vectors that rep- resent real-world objects? Quite a lot, it turns out. One use case is the topic of this monograph: the fundamental problem of retrieval. | 2401.09350#50 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 50,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We should be able to make similar arguments given a semantic embedding of text documents. Again consider the sparse embeddings with d being the size of the vocabulary, and more concretely, take SPLADE [Formal et al., 2021] as a concrete example. This model produces real-valued sparse vectors in an inner product space. In other words, the objective of its learning procedure is to maximize the inner product between similar vectors, where the inner product between two vectors u and v is denoted by (u,v) and is computed using >>, uri.\nIn the resulting space, if u, v, and w are generated by Splade with the property that â¨u, vâ© > â¨u, wâ©, then we can conclude that, according to Splade, documents represented by u and v are semantically more similar to each other than u is to w. There are numerous other examples of models that optimize for the angular distance or Euclidean distance (L2) between vectors to preserve (semantic) similarity.\nWhat can we do with a well-characterized collection of vectors that rep- resent real-world objects? Quite a lot, it turns out. One use case is the topic of this monograph: the fundamental problem of retrieval.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 51 | We are often interested in finding k objects that have the highest degree of similarity to a query object. When those objects are represented by vectors in a collection X , where the distance function δ(·, ·) is reflective of similarity, we may formalize this top-k question mathematically as finding the k minimizers of distance with the query point!
We state that formally in the following definition:
Definition 1.1 (Top-k Retrieval) Given a distance function δ(·, ·), we wish to pre-process a collection of data points X â Rd in time that is poly- nomial in |X | and d, to form a data structure (the âindexâ) whose size is
1.3 Flavors of Vector Retrieval
polynomial in |X | and d, so as to efficiently solve the following in time o(|X |d) for an arbitrary query q â Rd:
(k) arg min uâX δ(q, u). (1.1) | 2401.09350#51 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 51,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We are often interested in finding k objects that have the highest degree of similarity to a query object. When those objects are represented by vectors in a collection X , where the distance function δ(·, ·) is reflective of similarity, we may formalize this top-k question mathematically as finding the k minimizers of distance with the query point!\nWe state that formally in the following definition:\nDefinition 1.1 (Top-k Retrieval) Given a distance function δ(·, ·), we wish to pre-process a collection of data points X â Rd in time that is poly- nomial in |X | and d, to form a data structure (the âindexâ) whose size is\n1.3 Flavors of Vector Retrieval\npolynomial in |X | and d, so as to efficiently solve the following in time o(|X |d) for an arbitrary query q â Rd:\n(k) arg min uâX δ(q, u). (1.1)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 52 | (k) arg min uâX δ(q, u). (1.1)
A web search engine, for example, finds the most relevant documents to your query by first formulating it as a top-k retrieval problem over a collection of (not necessarily text-based) vectors. In this way, it quickly finds the subset of documents from the entire web that may satisfy the information need cap- tured in your query. Question answering systems, conversational agents (such as Siri, Alexa, and ChatGPT), recommendation engines, image search, out- lier detectors, and myriad other applications that are at the forefront of many online services and in many consumer gadgets depend on data structures and algorithms that can answer the top-k retrieval question as efficiently and as effectively as possible.
# 1.3 Flavors of Vector Retrieval
We create an instance of the deceptively simple problem formalized in Def- inition 1.1 the moment we acquire a collection of vectors X together with a distance function δ. In the remainder of this monograph, we assume that there is some function, either manually engineered or learnt, that transforms objects into vectors. So, from now on, X is a given.
The distance function then, specifies the flavor of the top-k retrieval prob- lem we need to solve. We will review these variations and explore what each entails.
# 1.3.1 Nearest Neighbor Search | 2401.09350#52 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 52,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "(k) arg min uâX δ(q, u). (1.1)\nA web search engine, for example, finds the most relevant documents to your query by first formulating it as a top-k retrieval problem over a collection of (not necessarily text-based) vectors. In this way, it quickly finds the subset of documents from the entire web that may satisfy the information need cap- tured in your query. Question answering systems, conversational agents (such as Siri, Alexa, and ChatGPT), recommendation engines, image search, out- lier detectors, and myriad other applications that are at the forefront of many online services and in many consumer gadgets depend on data structures and algorithms that can answer the top-k retrieval question as efficiently and as effectively as possible.\n# 1.3 Flavors of Vector Retrieval\nWe create an instance of the deceptively simple problem formalized in Def- inition 1.1 the moment we acquire a collection of vectors X together with a distance function δ. In the remainder of this monograph, we assume that there is some function, either manually engineered or learnt, that transforms objects into vectors. So, from now on, X is a given.\nThe distance function then, specifies the flavor of the top-k retrieval prob- lem we need to solve. We will review these variations and explore what each entails.\n# 1.3.1 Nearest Neighbor Search",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 53 | # 1.3.1 Nearest Neighbor Search
In many cases, the distance function is derived from a proper metric where non-negativity, coincidence, symmetry, and triangle inequality hold for δ. A clear example of this is the L2 distance: δ(u, v) = â¥u â vâ¥2. The resulting problem, illustrated for a toy example in Figure 1.2(a), is known as k-Nearest Neighbors (k-NN) search:
(k) arg min uâX â¥q â uâ¥2 = (k) arg min uâX â¥q â uâ¥2 2. (1.2)
7
8
1 Vector Retrieval | 2401.09350#53 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 53,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 1.3.1 Nearest Neighbor Search\nIn many cases, the distance function is derived from a proper metric where non-negativity, coincidence, symmetry, and triangle inequality hold for δ. A clear example of this is the L2 distance: δ(u, v) = â¥u â vâ¥2. The resulting problem, illustrated for a toy example in Figure 1.2(a), is known as k-Nearest Neighbors (k-NN) search:\n(k) arg min uâX â¥q â uâ¥2 = (k) arg min uâX â¥q â uâ¥2 2. (1.2)\n7\n8\n1 Vector Retrieval",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 55 | Fig. 1.2: Variants of vector retrieval for a toy vector collection in R2. In Nearest Neighbor search, we find the data point whose L2 distance to the query point is minimal (v for top-1 search). In Maximum Cosine Similarity search, we instead find the point whose angular distance to the query point is minimal (v and p are equidistant from the query). In Maximum Inner Product Search, we find a vector that maximizes the inner product with the query vector. This can be understood as letting the hyperplane orthogonal to the query point sweep the space towards the origin; the first vector to touch the sweeping plane is the maximizer of inner product. Another interpretation is this: the shaded region in the figure contains all the points y for which p is the answer to arg maxxâ{u,v,w,p}â¨x, yâ©.
# 1.3.2 Maximum Cosine Similarity Search
The distance function may also be the angular distance between vectors, which is again a proper metric. The resulting minimization problem can be stated as follows, though its equivalent maximization problem (involving the cosine of the angle between vectors) is perhaps more recognizable: | 2401.09350#55 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 55,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 1.2: Variants of vector retrieval for a toy vector collection in R2. In Nearest Neighbor search, we find the data point whose L2 distance to the query point is minimal (v for top-1 search). In Maximum Cosine Similarity search, we instead find the point whose angular distance to the query point is minimal (v and p are equidistant from the query). In Maximum Inner Product Search, we find a vector that maximizes the inner product with the query vector. This can be understood as letting the hyperplane orthogonal to the query point sweep the space towards the origin; the first vector to touch the sweeping plane is the maximizer of inner product. Another interpretation is this: the shaded region in the figure contains all the points y for which p is the answer to arg maxxâ{u,v,w,p}â¨x, yâ©.\n# 1.3.2 Maximum Cosine Similarity Search\nThe distance function may also be the angular distance between vectors, which is again a proper metric. The resulting minimization problem can be stated as follows, though its equivalent maximization problem (involving the cosine of the angle between vectors) is perhaps more recognizable:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 56 | (k) arg min uâX 1 â â¨q, uâ© â¥qâ¥2â¥uâ¥2 = (k) arg max uâX â¨q, uâ© â¥uâ¥2 . (1.3)
The latter is referred to as the k-Maximum Cosine Similarity (k-MCS) prob- lem. Note that, because the norm of the query point, â¥qâ¥2, is a constant in the optimization problem, it can simply be discarded; the resulting distance function is rank-equivalent to the angular distance. Figure 1.2(b) visualizes this problem on a toy collection of vectors.
1.3 Flavors of Vector Retrieval
# 1.3.3 Maximum Inner Product Search
Both of the problems in Equations (1.2) and (1.3) are special instances of a more general problem known as k-Maximum Inner Product Search (k-MIPS):
(k) arg max uâX â¨q, uâ©. (1.4) | 2401.09350#56 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 56,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "(k) arg min uâX 1 â â¨q, uâ© â¥qâ¥2â¥uâ¥2 = (k) arg max uâX â¨q, uâ© â¥uâ¥2 . (1.3)\nThe latter is referred to as the k-Maximum Cosine Similarity (k-MCS) prob- lem. Note that, because the norm of the query point, â¥qâ¥2, is a constant in the optimization problem, it can simply be discarded; the resulting distance function is rank-equivalent to the angular distance. Figure 1.2(b) visualizes this problem on a toy collection of vectors.\n1.3 Flavors of Vector Retrieval\n# 1.3.3 Maximum Inner Product Search\nBoth of the problems in Equations (1.2) and (1.3) are special instances of a more general problem known as k-Maximum Inner Product Search (k-MIPS):\n(k) arg max uâX â¨q, uâ©. (1.4)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 57 | (k) arg max uâX â¨q, uâ©. (1.4)
This is easy to see for k-MCS: If, in a pre-processing step, we L2-normalized all vectors in X so that u is transformed to uâ² = u/â¥uâ¥2, then â¥uâ²â¥2 = 1 and therefore Equation (1.3) reduces to Equation (1.4).
As for a reduction of k-NN to k-MIPS, we can expand Equation (1.2) as follows:
(k) arg min uâX â¥q â uâ¥2 2 = (k) arg min uâX â¥qâ¥2 2 â 2â¨q, uâ© + â¥uâ¥2 2 = (k) arg max uâ²âX â² â¨qâ², uâ²â©,
2, and defined qâ² â Rd+1 as where we have discarded the constant term, â¥qâ¥2 the concatenation of q â Rd and a 1-dimensional vector with value â1/2 (i.e., qâ² = [q, â1/2]), and uâ² â Rd+1 as [u, â¥uâ¥2 | 2401.09350#57 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 57,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "(k) arg max uâX â¨q, uâ©. (1.4)\nThis is easy to see for k-MCS: If, in a pre-processing step, we L2-normalized all vectors in X so that u is transformed to uâ² = u/â¥uâ¥2, then â¥uâ²â¥2 = 1 and therefore Equation (1.3) reduces to Equation (1.4).\nAs for a reduction of k-NN to k-MIPS, we can expand Equation (1.2) as follows:\n(k) arg min uâX â¥q â uâ¥2 2 = (k) arg min uâX â¥qâ¥2 2 â 2â¨q, uâ© + â¥uâ¥2 2 = (k) arg max uâ²âX â² â¨qâ², uâ²â©,\n2, and defined qâ² â Rd+1 as where we have discarded the constant term, â¥qâ¥2 the concatenation of q â Rd and a 1-dimensional vector with value â1/2 (i.e., qâ² = [q, â1/2]), and uâ² â Rd+1 as [u, â¥uâ¥2",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 58 | The k-MIPS problem, illustrated on a toy collection in Figure 1.2(c), does not come about just as the result of the reductions shown above. In fact, there exist embedding models (such as Splade, as discussed earlier) that learn vector representations with respect to inner product as the distance function. In other words, k-MIPS is an important problem in its own right.
# 1.3.3.1 Properties of MIPS
In a sense, then, it is sufficient to solve the k-MIPS problem as it is the umbrella problem for much of vector retrieval. Unfortunately, k-MIPS is a much harder problem than the other variants. That is because inner product is not a proper metric. In particular, it is not non-negative and does not satisfy the triangle inequality, so that â¨u, vâ© â® â¨u, wâ© + â¨w, vâ© in general.
Perhaps more troubling is the fact that even âcoincidenceâ is not guar- anteed. In other words, it is not true in general that a vector u maximizes inner product with itself: u ̸= arg maxvâX â¨v, uâ©! | 2401.09350#58 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 58,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The k-MIPS problem, illustrated on a toy collection in Figure 1.2(c), does not come about just as the result of the reductions shown above. In fact, there exist embedding models (such as Splade, as discussed earlier) that learn vector representations with respect to inner product as the distance function. In other words, k-MIPS is an important problem in its own right.\n# 1.3.3.1 Properties of MIPS\nIn a sense, then, it is sufficient to solve the k-MIPS problem as it is the umbrella problem for much of vector retrieval. Unfortunately, k-MIPS is a much harder problem than the other variants. That is because inner product is not a proper metric. In particular, it is not non-negative and does not satisfy the triangle inequality, so that â¨u, vâ© â® â¨u, wâ© + â¨w, vâ© in general.\nPerhaps more troubling is the fact that even âcoincidenceâ is not guar- anteed. In other words, it is not true in general that a vector u maximizes inner product with itself: u ̸= arg maxvâX â¨v, uâ©!",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 59 | As an example, suppose v and p = αv for some α > 1 are vectors in the collection X âa case demonstrated in Figure 1.2(c). Clearly, we have that
9
10
1 Vector Retrieval
â¨v, pâ© = αâ¨v, vâ© > â¨v, vâ©, so that p (and not v) is the solution to MIPS1 for the query point v.
In high-enough dimensions and under certain statistical conditions, how- ever, coincidence is reinstated for MIPS with high probability. One such case is stated in the following theorem.
Theorem 1.1 Suppose data points X are independent and identically dis- tributed ( iid) in each dimension and drawn from a zero-mean distribution. Then, for any u â X :
lim P [u = argmax(u, v)] = 1. doo VEX | 2401.09350#59 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 59,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "As an example, suppose v and p = αv for some α > 1 are vectors in the collection X âa case demonstrated in Figure 1.2(c). Clearly, we have that\n9\n10\n1 Vector Retrieval\nâ¨v, pâ© = αâ¨v, vâ© > â¨v, vâ©, so that p (and not v) is the solution to MIPS1 for the query point v.\nIn high-enough dimensions and under certain statistical conditions, how- ever, coincidence is reinstated for MIPS with high probability. One such case is stated in the following theorem.\nTheorem 1.1 Suppose data points X are independent and identically dis- tributed ( iid) in each dimension and drawn from a zero-mean distribution. Then, for any u â X :\nlim P [u = argmax(u, v)] = 1. doo VEX",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 60 | lim P [u = argmax(u, v)] = 1. doo VEX
Proof. Denote by Var[·] and E[·] the variance and expected value operators. By the conditions of the theorem, it is clear that E[â¨u, uâ©] = d E[Z 2] where Z is the random variable that generates each coordinate of the vector. We can also see that E[â¨u, Xâ©] = 0 for a random data point X, and that Var[â¨u, Xâ©] = â¥uâ¥2 2 We wish to claim that u â X is the solution to a MIPS problem where u is also the query point. That happens if and only if every other vector in X has an inner product with u that is smaller than â¨u, uâ©. So that:
P [wu = arg max(u,v)] =P [(u,v) < (uu) Vue a} = VEX 1-P[Svea st. (u,v) > (u,u)] > (by Union Bound) 1- Ss P[(u,v) > (u,u)] = (by id) vex 1-|&|P[(u,X) > (u,u)]. | 2401.09350#60 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 60,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "lim P [u = argmax(u, v)] = 1. doo VEX\nProof. Denote by Var[·] and E[·] the variance and expected value operators. By the conditions of the theorem, it is clear that E[â¨u, uâ©] = d E[Z 2] where Z is the random variable that generates each coordinate of the vector. We can also see that E[â¨u, Xâ©] = 0 for a random data point X, and that Var[â¨u, Xâ©] = â¥uâ¥2 2 We wish to claim that u â X is the solution to a MIPS problem where u is also the query point. That happens if and only if every other vector in X has an inner product with u that is smaller than â¨u, uâ©. So that:\nP [wu = arg max(u,v)] =P [(u,v) < (uu) Vue a} = VEX 1-P[Svea st. (u,v) > (u,u)] > (by Union Bound) 1- Ss P[(u,v) > (u,u)] = (by id) vex 1-|&|P[(u,X) > (u,u)].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 61 | Let us turn to the last term and bound the probability for a random data point:
P [(u,X) > (u,u)] =P [ (u, X) â (u,u) + dE[Z?] > dE[Z?]}. Y
The expected value of Y is 0. Denote by Ï2 its variance. By the application of the one-sided Chebyshevâs inequality,2 we arrive at the following bound:
2 P [lu X) > (0) < rapa
1 When k = 1, we drop the symbol k from the name of the retrieval problem. So we write MIPS instead of 1-MIPS. ? The one-sided Chebyshevâs inequality for a random variable X with mean p and variance o states that P [x âp> t] < a? /(0? + 0).
1.3 Flavors of Vector Retrieval
11
1.0f/ sw. : ¢ --©--Exp(1) : ia o-u(â-VB/2, VB/2) if â> U(0, 12) 10! 10? DIMENSIONALITY (d) | 2401.09350#61 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 61,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Let us turn to the last term and bound the probability for a random data point:\nP [(u,X) > (u,u)] =P [ (u, X) â (u,u) + dE[Z?] > dE[Z?]}. Y\nThe expected value of Y is 0. Denote by Ï2 its variance. By the application of the one-sided Chebyshevâs inequality,2 we arrive at the following bound:\n2 P [lu X) > (0) < rapa\n1 When k = 1, we drop the symbol k from the name of the retrieval problem. So we write MIPS instead of 1-MIPS. ? The one-sided Chebyshevâs inequality for a random variable X with mean p and variance o states that P [x âp> t] < a? /(0? + 0).\n1.3 Flavors of Vector Retrieval\n11\n1.0f/ sw. : ¢ --©--Exp(1) : ia o-u(â-VB/2, VB/2) if â> U(0, 12) 10! 10? DIMENSIONALITY (d)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 62 | Frver-TasB Ee . NQ-ESPIADE + NQ-TasB GLoVp-200. QuoraâSpLAaDE GioVE-100 ° > Quora-MintLM Quora-ESPLADE FEVER-MINILM . GLoYE-50 ° 2.0. NQ-MintLM GI OVE-25 0.0 : 102 10° 104 DIMENSIONALITY (d)
(a) Synthetic (b) Real
Fig. 1.3: Probability that u â X is the solution to MIPS over X with query u versus the dimensionality d, for various synthetic and real collections X . For synthetic collections, |X | = 100,000. Appendix A gives a description of the real collections. Note that, for real collections, we estimate the reported probability by sampling 10,000 data points and using them as queries. Fur- thermore, we do not pre-process the vectorsâimportantly, we do not L2- normalize the collections.
Note that, 0? is a function of the sum of iid random variables, and, as such, grows linearly with d. In the limit. this probability tends to 0. We have thus shown that limg.. P [u = arg max,¢(u,v)] > 1 which concludes the proof. Oo | 2401.09350#62 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 62,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Frver-TasB Ee . NQ-ESPIADE + NQ-TasB GLoVp-200. QuoraâSpLAaDE GioVE-100 ° > Quora-MintLM Quora-ESPLADE FEVER-MINILM . GLoYE-50 ° 2.0. NQ-MintLM GI OVE-25 0.0 : 102 10° 104 DIMENSIONALITY (d)\n(a) Synthetic (b) Real \nFig. 1.3: Probability that u â X is the solution to MIPS over X with query u versus the dimensionality d, for various synthetic and real collections X . For synthetic collections, |X | = 100,000. Appendix A gives a description of the real collections. Note that, for real collections, we estimate the reported probability by sampling 10,000 data points and using them as queries. Fur- thermore, we do not pre-process the vectorsâimportantly, we do not L2- normalize the collections.\nNote that, 0? is a function of the sum of iid random variables, and, as such, grows linearly with d. In the limit. this probability tends to 0. We have thus shown that limg.. P [u = arg max,¢(u,v)] > 1 which concludes the proof. Oo",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 63 | # 1.3.3.2 Empirical Demonstration of the Lack of Coincidence
Let us demonstrate the effect of Theorem 1.1 empirically. First, let us choose distributions that meet the requirements of the theorem: a Gaussian dis- tribution with mean 0 and variance 1, and a uniform distribution over [â 12/2] (with variance 1) will do. For comparison, choose another set of distributions that do not have the requisite properties: Exponential 12]. Having fixed the distributions, we with rate 1 and uniform over [0, next sample 100,000 random vectors from them to form a collection X . We then take each data point, use it as a query in MIPS over X , and report the proportion of data points that are solutions to their own search.
Figure 1.3(a) illustrates the results of this experiment. As expected, for the Gaussian and centered uniform distributions, the ratio of interest approaches 1 when d is sufficiently large. Surprisingly, even when the distributions do not strictly satisfy the conditions of the theorem, we still observe the convergence
12
1 Vector Retrieval
(a) k-NN (b) k-MCS (c) k-MIPS | 2401.09350#63 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 63,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 1.3.3.2 Empirical Demonstration of the Lack of Coincidence\nLet us demonstrate the effect of Theorem 1.1 empirically. First, let us choose distributions that meet the requirements of the theorem: a Gaussian dis- tribution with mean 0 and variance 1, and a uniform distribution over [â 12/2] (with variance 1) will do. For comparison, choose another set of distributions that do not have the requisite properties: Exponential 12]. Having fixed the distributions, we with rate 1 and uniform over [0, next sample 100,000 random vectors from them to form a collection X . We then take each data point, use it as a query in MIPS over X , and report the proportion of data points that are solutions to their own search.\nFigure 1.3(a) illustrates the results of this experiment. As expected, for the Gaussian and centered uniform distributions, the ratio of interest approaches 1 when d is sufficiently large. Surprisingly, even when the distributions do not strictly satisfy the conditions of the theorem, we still observe the convergence\n12\n1 Vector Retrieval\n(a) k-NN (b) k-MCS (c) k-MIPS",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 64 | Fig. 1.4: Approximate variants of top-1 retrieval for a toy collection in R2. In NN, we admit vectors that are at most ϵ away from the optimal solution. As such, x and y are both valid solutions as they are in a ball with radius (1 + ϵ)δ(q, x) centered at the query. Similarly, in MCS, we accept a vector (e.g., x) if its angle with the query point is at most 1+ϵ greater than the angle between the query and the optimal vector (i.e., v). For the MIPS example, assuming that the inner product of query and x is at most (1 â ϵ)-times the inner product of query and p, then x is an acceptable solution.
of that ratio to 1. So it appears that the requirements of Theorem 1.1 are more forgiving than one may imagine. | 2401.09350#64 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 64,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 1.4: Approximate variants of top-1 retrieval for a toy collection in R2. In NN, we admit vectors that are at most ϵ away from the optimal solution. As such, x and y are both valid solutions as they are in a ball with radius (1 + ϵ)δ(q, x) centered at the query. Similarly, in MCS, we accept a vector (e.g., x) if its angle with the query point is at most 1+ϵ greater than the angle between the query and the optimal vector (i.e., v). For the MIPS example, assuming that the inner product of query and x is at most (1 â ϵ)-times the inner product of query and p, then x is an acceptable solution.\nof that ratio to 1. So it appears that the requirements of Theorem 1.1 are more forgiving than one may imagine.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 65 | of that ratio to 1. So it appears that the requirements of Theorem 1.1 are more forgiving than one may imagine.
We also repeat the exercise above on several real-world collections, a de- scription of which can be found in Appendix A along with salient statistics. The results of these experiments are visualized in Figure 1.3(b). As expected, whether a data point maximizes inner product with itself entirely depends on the underlying data distribution. We can observe that, for some collections in high dimensions, we are likely to encounter coincidence in the sense we defined earlier, but for others that is clearly not the case. It is important to keep this difference between synthetic and real collections in mind when designing experiments that evaluate the performance of MIPS systems.
# 1.4 Approximate Vector Retrieval
Saying one problem is harder than another neither implies that we cannot approach the harder problem, nor does it mean that the âeasierâ problem is easy to solve. In fact, none of these variants of vector retrieval (k-NN, k- MCS, and k-MIPS) can be solved exactly and efficiently in high dimensions. Instead, we must either accept that the solution would be inefficient (in terms of space- or time-complexity), or allow some degree of error.
1.4 Approximate Vector Retrieval | 2401.09350#65 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 65,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "of that ratio to 1. So it appears that the requirements of Theorem 1.1 are more forgiving than one may imagine.\nWe also repeat the exercise above on several real-world collections, a de- scription of which can be found in Appendix A along with salient statistics. The results of these experiments are visualized in Figure 1.3(b). As expected, whether a data point maximizes inner product with itself entirely depends on the underlying data distribution. We can observe that, for some collections in high dimensions, we are likely to encounter coincidence in the sense we defined earlier, but for others that is clearly not the case. It is important to keep this difference between synthetic and real collections in mind when designing experiments that evaluate the performance of MIPS systems.\n# 1.4 Approximate Vector Retrieval\nSaying one problem is harder than another neither implies that we cannot approach the harder problem, nor does it mean that the âeasierâ problem is easy to solve. In fact, none of these variants of vector retrieval (k-NN, k- MCS, and k-MIPS) can be solved exactly and efficiently in high dimensions. Instead, we must either accept that the solution would be inefficient (in terms of space- or time-complexity), or allow some degree of error.\n1.4 Approximate Vector Retrieval",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 66 | 1.4 Approximate Vector Retrieval
The first case of solving the problem exactly but inefficiently is uninterest- ing: If we are looking to find the solution for k = 1, for example, it is enough to compute the distance function for every vector in the collection and the query, resulting in linear complexity. When k > 1, the total time complexity is O(|X |d log k), where |X | is the size of the collection. So it typically makes more sense to investigate the second strategy of admitting error.
That argument leads naturally to the class of ϵ-approximate vector re- trieval problems. This idea can be formalized rather easily for the special case where k = 1: The approximate solution for the top-1 retrieval is satis- factory so long as the vector u returned by the algorithm is at most (1 + ϵ) factor farther than the optimal vector uâ, according to δ(·, ·) and for some arbitrary ϵ > 0:
δ(q, u) ⤠(1 + ϵ)δ(q, uâ). (1.5)
# Figure 1.4 renders the solution space for an example collection in R2. | 2401.09350#66 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 66,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "1.4 Approximate Vector Retrieval\nThe first case of solving the problem exactly but inefficiently is uninterest- ing: If we are looking to find the solution for k = 1, for example, it is enough to compute the distance function for every vector in the collection and the query, resulting in linear complexity. When k > 1, the total time complexity is O(|X |d log k), where |X | is the size of the collection. So it typically makes more sense to investigate the second strategy of admitting error.\nThat argument leads naturally to the class of ϵ-approximate vector re- trieval problems. This idea can be formalized rather easily for the special case where k = 1: The approximate solution for the top-1 retrieval is satis- factory so long as the vector u returned by the algorithm is at most (1 + ϵ) factor farther than the optimal vector uâ, according to δ(·, ·) and for some arbitrary ϵ > 0:\nδ(q, u) ⤠(1 + ϵ)δ(q, uâ). (1.5)\n# Figure 1.4 renders the solution space for an example collection in R2.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 67 | # Figure 1.4 renders the solution space for an example collection in R2.
The formalism above extends to the more general case where k > 1 in an obvious way: a vector u is a valid solution to the ϵ-approximate top-k problem if its distance to the query point is at most (1 + ϵ) times the distance to the k-th optimal vector. This is summarized in the following definition:
Given a distance Definition 1.2 (ϵ-Approximate Top-k Retrieval) function δ(·, ·), we wish to pre-process a collection of data points X â Rd in time that is polynomial in |X | and d, to form a data structure (the âin- dexâ) whose size is polynomial in |X | and d, so as to efficiently solve the following in time o(|X |d) for an arbitrary query q â Rd and ϵ > 0:
S = (k) arg min uâX δ(q, u),
such that for all u â S, Equation (1.5) is satisfied where uâ is the k-th optimal vector obtained by solving the problem in Definition 1.1. | 2401.09350#67 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 67,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# Figure 1.4 renders the solution space for an example collection in R2.\nThe formalism above extends to the more general case where k > 1 in an obvious way: a vector u is a valid solution to the ϵ-approximate top-k problem if its distance to the query point is at most (1 + ϵ) times the distance to the k-th optimal vector. This is summarized in the following definition:\nGiven a distance Definition 1.2 (ϵ-Approximate Top-k Retrieval) function δ(·, ·), we wish to pre-process a collection of data points X â Rd in time that is polynomial in |X | and d, to form a data structure (the âin- dexâ) whose size is polynomial in |X | and d, so as to efficiently solve the following in time o(|X |d) for an arbitrary query q â Rd and ϵ > 0:\nS = (k) arg min uâX δ(q, u),\nsuch that for all u â S, Equation (1.5) is satisfied where uâ is the k-th optimal vector obtained by solving the problem in Definition 1.1.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 68 | such that for all u â S, Equation (1.5) is satisfied where uâ is the k-th optimal vector obtained by solving the problem in Definition 1.1.
Despite the extension to top-k above, it is more common to characterize the effectiveness of an approximate top-k solution as the percentage of correct vectors that are present in the solution. Concretely, if S = arg max(k) uâX δ(q, u) is the exact set of top-k vectors, and ËS is the approximate set, then the accuracy of the approximate algorithm can be reported as |S â© ËS|/k.3
This monograph primarily studies the approximate4 retrieval problem. As such, while we state a retrieval problem using the arg max or arg min notation, we are generally only interested in approximate solutions to it.
3 This quantity is also known as recall number of vectors our algorithm recalls from the exact solution set. 4 We drop ϵ from the name when it is clear from context.
in the literature, because we are counting the
13
14
1 Vector Retrieval
# References
Y. Bai, X. Li, G. Wang, C. Zhang, L. Shang, J. Xu, Z. Wang, F. Wang, and Q. Liu. Sparterm: Learning term-based sparse representation for fast text retrieval, 2020. | 2401.09350#68 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 68,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "such that for all u â S, Equation (1.5) is satisfied where uâ is the k-th optimal vector obtained by solving the problem in Definition 1.1.\nDespite the extension to top-k above, it is more common to characterize the effectiveness of an approximate top-k solution as the percentage of correct vectors that are present in the solution. Concretely, if S = arg max(k) uâX δ(q, u) is the exact set of top-k vectors, and ËS is the approximate set, then the accuracy of the approximate algorithm can be reported as |S â© ËS|/k.3\nThis monograph primarily studies the approximate4 retrieval problem. As such, while we state a retrieval problem using the arg max or arg min notation, we are generally only interested in approximate solutions to it.\n3 This quantity is also known as recall number of vectors our algorithm recalls from the exact solution set. 4 We drop ϵ from the name when it is clear from context.\nin the literature, because we are counting the\n13\n14\n1 Vector Retrieval\n# References\nY. Bai, X. Li, G. Wang, C. Zhang, L. Shang, J. Xu, Z. Wang, F. Wang, and Q. Liu. Sparterm: Learning term-based sparse representation for fast text retrieval, 2020.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 69 | S. Bruch, S. Gai, and A. Ingber. An analysis of fusion functions for hybrid retrieval. ACM Transactions on Information Systems, 42(1), 8 2023.
T. Chen, M. Zhang, J. Lu, M. Bendersky, and M. Najork. Out-of-domain semantics to the rescue! zero-shot hybrid retrieval models. In Advances in Information Retrieval: 44th European Conference on IR Research, pages 95â110, 2022.
Z. Dai and J. Callan. Context-aware term weighting for first stage passage retrieval. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 1533â1536, 2020.
T. Formal, B. Piwowarski, and S. Clinchant. Splade: Sparse lexical and expan- sion model for first stage ranking. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Re- trieval, pages 2288â2292, 2021.
T. Formal, C. Lassance, B. Piwowarski, and S. Clinchant. From distillation to hard negative sampling: Making sparse neural ir models more effective. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 2353â2359, 2022. | 2401.09350#69 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 69,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Bruch, S. Gai, and A. Ingber. An analysis of fusion functions for hybrid retrieval. ACM Transactions on Information Systems, 42(1), 8 2023.\nT. Chen, M. Zhang, J. Lu, M. Bendersky, and M. Najork. Out-of-domain semantics to the rescue! zero-shot hybrid retrieval models. In Advances in Information Retrieval: 44th European Conference on IR Research, pages 95â110, 2022.\nZ. Dai and J. Callan. Context-aware term weighting for first stage passage retrieval. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 1533â1536, 2020.\nT. Formal, B. Piwowarski, and S. Clinchant. Splade: Sparse lexical and expan- sion model for first stage ranking. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Re- trieval, pages 2288â2292, 2021.\nT. Formal, C. Lassance, B. Piwowarski, and S. Clinchant. From distillation to hard negative sampling: Making sparse neural ir models more effective. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, page 2353â2359, 2022.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 70 | L. Gao, Z. Dai, and J. Callan. COIL: revisit exact lexical match in informa- tion retrieval with contextualized inverted list. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Compu- tational Linguistics: Human Language Technologies, NAACL-HLT 2021, Online, June 6-11, 2021, pages 3030â3042, 2021.
W. Guo, J. Wang, and S. Wang. Deep multimodal representation learning: A survey. IEEE Access, 7:63373â63394, 2019.
V. Karpukhin, B. Oguz, S. Min, P. Lewis, L. Wu, S. Edunov, D. Chen, and W.-t. Yih. Dense passage retrieval for open-domain question answering. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing, pages 6769â6781, Nov. 2020.
O. Khattab and M. Zaharia. Colbert: Efficient and effective passage search via contextualized late interaction over bert. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 39â48, 2020. | 2401.09350#70 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 70,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "L. Gao, Z. Dai, and J. Callan. COIL: revisit exact lexical match in informa- tion retrieval with contextualized inverted list. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Compu- tational Linguistics: Human Language Technologies, NAACL-HLT 2021, Online, June 6-11, 2021, pages 3030â3042, 2021.\nW. Guo, J. Wang, and S. Wang. Deep multimodal representation learning: A survey. IEEE Access, 7:63373â63394, 2019.\nV. Karpukhin, B. Oguz, S. Min, P. Lewis, L. Wu, S. Edunov, D. Chen, and W.-t. Yih. Dense passage retrieval for open-domain question answering. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing, pages 6769â6781, Nov. 2020.\nO. Khattab and M. Zaharia. Colbert: Efficient and effective passage search via contextualized late interaction over bert. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 39â48, 2020.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 71 | S. Kuzi, M. Zhang, C. Li, M. Bendersky, and M. Najork. Leveraging semantic and lexical matching to improve the recall of document retrieval systems: A hybrid approach, 2020.
J. Lin and X. Ma. A few brief notes on deepimpact, coil, and a conceptual framework for information retrieval techniques, 2021.
J. Lin, R. Nogueira, and A. Yates. Pretrained Transformers for Text Ranking: BERT and Beyond. Springer Cham, 2021.
References
J. Ma, I. Korotkov, K. Hall, and R. T. McDonald. Hybrid first-stage retrieval models for biomedical literature. In CLEF, 2020.
X. Ma, K. Sun, R. Pradeep, and J. J. Lin. A replication study of dense passage retriever, 2021.
A. Mallia, O. Khattab, T. Suel, and N. Tonellotto. Learning passage impacts for inverted indexes. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 1723â1727, 2021. | 2401.09350#71 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 71,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Kuzi, M. Zhang, C. Li, M. Bendersky, and M. Najork. Leveraging semantic and lexical matching to improve the recall of document retrieval systems: A hybrid approach, 2020.\nJ. Lin and X. Ma. A few brief notes on deepimpact, coil, and a conceptual framework for information retrieval techniques, 2021.\nJ. Lin, R. Nogueira, and A. Yates. Pretrained Transformers for Text Ranking: BERT and Beyond. Springer Cham, 2021.\nReferences\nJ. Ma, I. Korotkov, K. Hall, and R. T. McDonald. Hybrid first-stage retrieval models for biomedical literature. In CLEF, 2020.\nX. Ma, K. Sun, R. Pradeep, and J. J. Lin. A replication study of dense passage retriever, 2021.\nA. Mallia, O. Khattab, T. Suel, and N. Tonellotto. Learning passage impacts for inverted indexes. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 1723â1727, 2021.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 72 | N. Reimers and I. Gurevych. Sentence-bert: Sentence embeddings using siamese bert-networks. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, 11 2019.
G. Salton and C. Buckley. Term-weighting approaches in automatic text retrieval. Information Processing and Management, 24(5):513â523, 1988. K. Santhanam, O. Khattab, J. Saad-Falcon, C. Potts, and M. Zaharia. Col- BERTv2: Effective and efficient retrieval via lightweight late interaction. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 3715â3734, July 2022.
A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems, page 6000â6010, 2017. | 2401.09350#72 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 72,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "N. Reimers and I. Gurevych. Sentence-bert: Sentence embeddings using siamese bert-networks. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, 11 2019.\nG. Salton and C. Buckley. Term-weighting approaches in automatic text retrieval. Information Processing and Management, 24(5):513â523, 1988. K. Santhanam, O. Khattab, J. Saad-Falcon, C. Potts, and M. Zaharia. Col- BERTv2: Effective and efficient retrieval via lightweight late interaction. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 3715â3734, July 2022.\nA. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems, page 6000â6010, 2017.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 73 | S. Wang, S. Zhuang, and G. Zuccon. Bert-based dense retrievers require interpolation with bm25 for effective passage retrieval. In Proceedings of the 2021 ACM SIGIR International Conference on Theory of Information Retrieval, page 317â324, 2021.
X. Wu, R. Guo, D. Simcha, D. Dopson, and S. Kumar. Efficient inner product approximation in hybrid spaces, 2019.
L. Xiong, C. Xiong, Y. Li, K.-F. Tang, J. Liu, P. Bennett, J. Ahmed, and A. Overwijk. Approximate nearest neighbor negative contrastive learning for dense text retrieval. In International Conference on Learning Repre- sentations, 4 2021.
H. Zamani, M. Dehghani, W. B. Croft, E. Learned-Miller, and J. Kamps. From neural re-ranking to neural ranking: Learning a sparse representa- tion for inverted indexing. In Proceedings of the 27th ACM International Conference on Information and Knowledge Management, pages 497â506, 2018.
C. Zhang, Z. Yang, X. He, and L. Deng. Multimodal intelligence: Repre- sentation learning, information fusion, and applications. IEEE Journal of Selected Topics in Signal Processing, 14(3):478â493, 2020. | 2401.09350#73 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 73,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Wang, S. Zhuang, and G. Zuccon. Bert-based dense retrievers require interpolation with bm25 for effective passage retrieval. In Proceedings of the 2021 ACM SIGIR International Conference on Theory of Information Retrieval, page 317â324, 2021.\nX. Wu, R. Guo, D. Simcha, D. Dopson, and S. Kumar. Efficient inner product approximation in hybrid spaces, 2019.\nL. Xiong, C. Xiong, Y. Li, K.-F. Tang, J. Liu, P. Bennett, J. Ahmed, and A. Overwijk. Approximate nearest neighbor negative contrastive learning for dense text retrieval. In International Conference on Learning Repre- sentations, 4 2021.\nH. Zamani, M. Dehghani, W. B. Croft, E. Learned-Miller, and J. Kamps. From neural re-ranking to neural ranking: Learning a sparse representa- tion for inverted indexing. In Proceedings of the 27th ACM International Conference on Information and Knowledge Management, pages 497â506, 2018.\nC. Zhang, Z. Yang, X. He, and L. Deng. Multimodal intelligence: Repre- sentation learning, information fusion, and applications. IEEE Journal of Selected Topics in Signal Processing, 14(3):478â493, 2020.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 74 | S. Zhuang and G. Zuccon. Fast passage re-ranking with contextualized exact term matching and efficient passage expansion. In Workshop on Reaching Efficiency in Neural Information Retrieval, the 45th International ACM
15
16
1 Vector Retrieval
SIGIR Conference on Research and Development in Information Retrieval, 2022.
# Chapter 2 Retrieval Stability in High Dimensions
Abstract We are about to embark on a comprehensive survey and analysis of vector retrieval methods in the remainder of this monograph. It may thus sound odd to suggest that you may not need any of these clever ideas in order to perform vector retrieval. Sometimes, under bizarrely general conditions that we will explore formally in this chapter, an exhaustive search (where we compute the distance between query and every data point, sort, and return the top k) is likely to perform much better in both accuracy and search latency! The reason why that may be the case has to do with the approximate nature of algorithms and the oddities of high dimensions. We elaborate this point by focusing on the top-1 case.
# 2.1 Intuition | 2401.09350#74 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 74,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Zhuang and G. Zuccon. Fast passage re-ranking with contextualized exact term matching and efficient passage expansion. In Workshop on Reaching Efficiency in Neural Information Retrieval, the 45th International ACM\n15\n16\n1 Vector Retrieval\nSIGIR Conference on Research and Development in Information Retrieval, 2022.\n# Chapter 2 Retrieval Stability in High Dimensions\nAbstract We are about to embark on a comprehensive survey and analysis of vector retrieval methods in the remainder of this monograph. It may thus sound odd to suggest that you may not need any of these clever ideas in order to perform vector retrieval. Sometimes, under bizarrely general conditions that we will explore formally in this chapter, an exhaustive search (where we compute the distance between query and every data point, sort, and return the top k) is likely to perform much better in both accuracy and search latency! The reason why that may be the case has to do with the approximate nature of algorithms and the oddities of high dimensions. We elaborate this point by focusing on the top-1 case.\n# 2.1 Intuition",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 75 | # 2.1 Intuition
Consider the case of proper distance functions where δ(·, ·) is a metric. Recall from Equation (1.5) that a vector u is an acceptable ϵ-approximate solution if its distance to the query q according to δ(·, ·) is at most (1 + ϵ)δ(q, uâ), where uâ is the optimal vector and ϵ is an arbitrary parameter. As shown in Figure 1.4(a) for NN, this means that, if you centered an Lp ball around q with radius δ(q, (1 + ϵ)uâ), then u is in that ball.
So, what if we find ourselves in a situation where no matter how small ϵ is, too many vectors, or indeed all vectors, from our collection X end up in the (1 + ϵ)-enlarged ball? Then, by definition, every vector is an ϵ-approximate nearest neighbor of q!
In such a configuration of points, it is questionable whether the notion of ânearest neighborâ has any meaning at all: If the query point were perturbed by some noise as small as ϵ, then its true nearest neighbor would suddenly change, making NN unstable. Because of that instability, any approximate algorithm will need to examine a large portion or nearly all of the data
17
18 | 2401.09350#75 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 75,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 2.1 Intuition\nConsider the case of proper distance functions where δ(·, ·) is a metric. Recall from Equation (1.5) that a vector u is an acceptable ϵ-approximate solution if its distance to the query q according to δ(·, ·) is at most (1 + ϵ)δ(q, uâ), where uâ is the optimal vector and ϵ is an arbitrary parameter. As shown in Figure 1.4(a) for NN, this means that, if you centered an Lp ball around q with radius δ(q, (1 + ϵ)uâ), then u is in that ball.\nSo, what if we find ourselves in a situation where no matter how small ϵ is, too many vectors, or indeed all vectors, from our collection X end up in the (1 + ϵ)-enlarged ball? Then, by definition, every vector is an ϵ-approximate nearest neighbor of q!\nIn such a configuration of points, it is questionable whether the notion of ânearest neighborâ has any meaning at all: If the query point were perturbed by some noise as small as ϵ, then its true nearest neighbor would suddenly change, making NN unstable. Because of that instability, any approximate algorithm will need to examine a large portion or nearly all of the data\n17\n18",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 76 | 17
18
2 Retrieval Stability in High Dimensions
points anyway, reducing thereby to a procedure that performs more poorly than exhaustive search.
That sounds troubling. But when might we experience that phenomenon? That is the question Beyer et al. [1999] investigate in their seminal paper.
It turns out, one scenario where vector retrieval becomes unstable as dimensionality d increases is if a) data points are iid in each dimension, b) query points are similarly drawn iid in each dimension, and c) query points are independent of data points. This includes many synthetic collections that are, even today, routinely but inappropriately used for evaluation purposes.
On the other hand, when data points form clusters and query points fall into these same clusters, then the (approximate) ânearest clusterâ problem is meaningfulâbut not necessarily the approximate NN problem. So while it makes sense to use approximate algorithms to obtain the nearest cluster, search within clusters may as well be exhaustive. This, as we will learn in Chapter 7, is the basis for a popular and effective class of vector retrieval algorithms on real collections.
# 2.2 Formal Results | 2401.09350#76 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 76,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "17\n18\n2 Retrieval Stability in High Dimensions\npoints anyway, reducing thereby to a procedure that performs more poorly than exhaustive search.\nThat sounds troubling. But when might we experience that phenomenon? That is the question Beyer et al. [1999] investigate in their seminal paper.\nIt turns out, one scenario where vector retrieval becomes unstable as dimensionality d increases is if a) data points are iid in each dimension, b) query points are similarly drawn iid in each dimension, and c) query points are independent of data points. This includes many synthetic collections that are, even today, routinely but inappropriately used for evaluation purposes.\nOn the other hand, when data points form clusters and query points fall into these same clusters, then the (approximate) ânearest clusterâ problem is meaningfulâbut not necessarily the approximate NN problem. So while it makes sense to use approximate algorithms to obtain the nearest cluster, search within clusters may as well be exhaustive. This, as we will learn in Chapter 7, is the basis for a popular and effective class of vector retrieval algorithms on real collections.\n# 2.2 Formal Results",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 77 | # 2.2 Formal Results
More generally, vector retrieval becomes unstable in high dimensions when the variance of the distance between query and data points grows substan- tially more slowly than its expected value. That makes sense. Intuitively, that means that more and more data points fall into the (1 + ϵ)-enlarged ball cen- tered at the query. This can be stated formally as the following theorem due to Beyer et al. [1999], extended to any general distance function δ(·, ·).
Theorem 2.1 Suppose m data points X â Rd are drawn iid from a data distribution and a query point q is drawn independent of data points from any distribution. Denote by X a random data point. If
Jim Var [5(q,X)]/E [6(q,.X)]â = 0,
then for any ⬠> 0, limaâoo P [5(q,X) < (1+ 6d(q, u*)| = 1, where u* is the vector closest to q. | 2401.09350#77 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 77,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 2.2 Formal Results\nMore generally, vector retrieval becomes unstable in high dimensions when the variance of the distance between query and data points grows substan- tially more slowly than its expected value. That makes sense. Intuitively, that means that more and more data points fall into the (1 + ϵ)-enlarged ball cen- tered at the query. This can be stated formally as the following theorem due to Beyer et al. [1999], extended to any general distance function δ(·, ·).\nTheorem 2.1 Suppose m data points X â Rd are drawn iid from a data distribution and a query point q is drawn independent of data points from any distribution. Denote by X a random data point. If\nJim Var [5(q,X)]/E [6(q,.X)]â = 0,\nthen for any ⬠> 0, limaâoo P [5(q,X) < (1+ 6d(q, u*)| = 1, where u* is the vector closest to q.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 78 | Proof. Let 5, = maxy,cx 6(q,u) and 6* = min,<x 6(q,u). If we could show that, for some d-dependent positive a and 8 such that B/a = 1+, limgoo P [a <<< 8] = 1, then we are done. That is because, in that case 6,/5* < 8/a = 1+ ⬠almost surely and the claim follows.
2.2 Formal Results
From the above, all that we need to do is to find α and β for a given d. Intuitively, we want the interval [α, β] to contain E[δ(q, X)], because we know from the condition of the theorem that the distances should concentrate around their mean. So α = (1 â η) E[δ(q, X)] and β = (1 + η) E[δ(q, X)] for some η seems like a reasonable choice. Letting η = ϵ/(ϵ + 2) gives us the desired ratio: β/α = 1 + ϵ.
Now we must show that δâ and δâ belong to our chosen [α, β] interval al- most surely in the limit. That happens if all distances belong to that interval. So: | 2401.09350#78 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 78,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. Let 5, = maxy,cx 6(q,u) and 6* = min,<x 6(q,u). If we could show that, for some d-dependent positive a and 8 such that B/a = 1+, limgoo P [a <<< 8] = 1, then we are done. That is because, in that case 6,/5* < 8/a = 1+ ⬠almost surely and the claim follows.\n2.2 Formal Results\nFrom the above, all that we need to do is to find α and β for a given d. Intuitively, we want the interval [α, β] to contain E[δ(q, X)], because we know from the condition of the theorem that the distances should concentrate around their mean. So α = (1 â η) E[δ(q, X)] and β = (1 + η) E[δ(q, X)] for some η seems like a reasonable choice. Letting η = ϵ/(ϵ + 2) gives us the desired ratio: β/α = 1 + ϵ.\nNow we must show that δâ and δâ belong to our chosen [α, β] interval al- most surely in the limit. That happens if all distances belong to that interval. So:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 79 | lim P [a < 6* <6, <6\= doo lim P [5(a, u) â¬[a,B] Yue x| = doo Jim P{(1ân) E[6(a, X)] < 6(g,u) < 0 +m) BIG, XI] Vue a] = Jim P {(6(a.w) â El6(4,X)]| < nE[g,X)] Vue 4]. 00
It is now easier to work with the complementary event:
1â lim P [3 ue Xs.t. |5(q,u) â E[6(q, X)]| > 7 B(6(4,X)]]. doo
Using the Union Bound, the probability above is greater than or equal to the following:
lim P [a < 6* <6. < S| > doo 1- jm nu? [|9(a, u) u) â E[6 (a, X)]| > 7B[5(a, X)]| = 1 Jim, ye [(0(a,w) â Bl6(a, X)))â > wP Bl6(a, XY]. | 2401.09350#79 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 79,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "lim P [a < 6* <6, <6\\= doo lim P [5(a, u) â¬[a,B] Yue x| = doo Jim P{(1ân) E[6(a, X)] < 6(g,u) < 0 +m) BIG, XI] Vue a] = Jim P {(6(a.w) â El6(4,X)]| < nE[g,X)] Vue 4]. 00\nIt is now easier to work with the complementary event:\n1â lim P [3 ue Xs.t. |5(q,u) â E[6(q, X)]| > 7 B(6(4,X)]]. doo\nUsing the Union Bound, the probability above is greater than or equal to the following:\nlim P [a < 6* <6. < S| > doo 1- jm nu? [|9(a, u) u) â E[6 (a, X)]| > 7B[5(a, X)]| = 1 Jim, ye [(0(a,w) â Bl6(a, X)))â > wP Bl6(a, XY].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 80 | Note that, q is independent of data points and that data points are iid ran- dom variables. Therefore, δ(q, u)âs are random variables drawn iid as well. Furthermore, by assumption E[δ(q, X)] exists, making it possible to apply Markovâs inequality to obtain the following bound:
lim P [a < 6* <6, < 6] > doo 1 Jim ||P [(3(g,X) ~ El6(4.X)))â > 0? E[o(g, XP] > doo 1â lim m 5 E|(5(q,u) â E[d(q, X)))] = Tee [(5(a,u) - B15(@, X)))"| 1 jm BIG
19
20
2 Retrieval Stability in High Dimensions
By the conditions of the theorem, Var[δ(q, X)]/ E[δ(q, X)]2 â 0 as d â â, so that the last expression tends to 1 in the limit. That concludes the proof. ââ | 2401.09350#80 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 80,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Note that, q is independent of data points and that data points are iid ran- dom variables. Therefore, δ(q, u)âs are random variables drawn iid as well. Furthermore, by assumption E[δ(q, X)] exists, making it possible to apply Markovâs inequality to obtain the following bound:\nlim P [a < 6* <6, < 6] > doo 1 Jim ||P [(3(g,X) ~ El6(4.X)))â > 0? E[o(g, XP] > doo 1â lim m 5 E|(5(q,u) â E[d(q, X)))] = Tee [(5(a,u) - B15(@, X)))\"| 1 jm BIG\n19\n20\n2 Retrieval Stability in High Dimensions\nBy the conditions of the theorem, Var[δ(q, X)]/ E[δ(q, X)]2 â 0 as d â â, so that the last expression tends to 1 in the limit. That concludes the proof. ââ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 81 | We mentioned earlier that if data and query points are independent of each other and that vectors are drawn iid in each dimension, then vector retrieval becomes unstable. For NN with the Z, norm, it is easy to show that such a configuration satisfies the conditions of Theorem 2.1, hence the instability. Consider the following for 6(q, u) = ||q â ull: , â ull? ; â~u,)P lim Var [||q wl fim Var [0 (ai wi)? So E[ligâulb) 4° ELL (a â ui)?
, â ull? ; â~u,)P lim Var [||q wl fim Var [0 (ai wi)? So E[ligâulb) 4° ELL (a â ui)? sag aN las 0") im +", dee (YE [(Gi â ui)?]) ki do? _ doÂ¥oe ape (by independence)
where we write Ï2 = Var[(qi â ui)p] and µ = E[(qi â ui)p].
When δ(q, u) = ââ¨q, uâ©, the same conditions result in retrieval instability: | 2401.09350#81 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 81,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We mentioned earlier that if data and query points are independent of each other and that vectors are drawn iid in each dimension, then vector retrieval becomes unstable. For NN with the Z, norm, it is easy to show that such a configuration satisfies the conditions of Theorem 2.1, hence the instability. Consider the following for 6(q, u) = ||q â ull: , â ull? ; â~u,)P lim Var [||q wl fim Var [0 (ai wi)? So E[ligâulb) 4° ELL (a â ui)?\n, â ull? ; â~u,)P lim Var [||q wl fim Var [0 (ai wi)? So E[ligâulb) 4° ELL (a â ui)? sag aN las 0\") im +\", dee (YE [(Gi â ui)?]) ki do? _ doÂ¥oe ape (by independence)\nwhere we write Ï2 = Var[(qi â ui)p] and µ = E[(qi â ui)p].\nWhen δ(q, u) = ââ¨q, uâ©, the same conditions result in retrieval instability:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 82 | When δ(q, u) = ââ¨q, uâ©, the same conditions result in retrieval instability:
Var [(q u)| _ Var [ Y: qui] im 3 lim 5 do El(qu)) 4° ELD, qui] li 2 Var [aie] (by independence) a (SO, E [aiui]) lo? Be
where we write Ï2 = Var[qiui] and µ = E[qiui].
# 2.3 Empirical Demonstration of Instability
Let us examine the theorem empirically. We simulate the NN setting with L2 distance and report the results in Figure 2.1. In these experiments, we sample 1,000,000 data points with each coordinate drawing its value independently from the same distribution, and 1,000 query points sampled similarly. We then compute the minimum and maximum distance between each query point and the data collection, measure the ratio between them, and report the mean and standard deviation of the ratio across queries. We repeat this exercise for various values of dimensionality d and render the results in Figure 2.1(a). Unsurprisingly, this ratio tends to 1 as d â â, as predicted by the theorem. Another way to understand this result is to count the number of data points that qualify as approximate nearest neighbors. The theory predicts
2.3 Empirical Demonstration of Instability
21 | 2401.09350#82 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 82,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "When δ(q, u) = ââ¨q, uâ©, the same conditions result in retrieval instability:\nVar [(q u)| _ Var [ Y: qui] im 3 lim 5 do El(qu)) 4° ELD, qui] li 2 Var [aie] (by independence) a (SO, E [aiui]) lo? Be\nwhere we write Ï2 = Var[qiui] and µ = E[qiui].\n# 2.3 Empirical Demonstration of Instability\nLet us examine the theorem empirically. We simulate the NN setting with L2 distance and report the results in Figure 2.1. In these experiments, we sample 1,000,000 data points with each coordinate drawing its value independently from the same distribution, and 1,000 query points sampled similarly. We then compute the minimum and maximum distance between each query point and the data collection, measure the ratio between them, and report the mean and standard deviation of the ratio across queries. We repeat this exercise for various values of dimensionality d and render the results in Figure 2.1(a). Unsurprisingly, this ratio tends to 1 as d â â, as predicted by the theorem. Another way to understand this result is to count the number of data points that qualify as approximate nearest neighbors. The theory predicts\n2.3 Empirical Demonstration of Instability\n21",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 83 | 2.3 Empirical Demonstration of Instability
21
G00 0.1) --©-Exp(1) â>-U(0, V2) 10! 10° DIMENSIONALITY (d)
100 iS} E s0 5 n t 60 B 3 = 40 5 a inf ->-64 & 20 âe128 < $256 3 0 12 0 25 50 75 100 ⬠AS PERCENT 6°
(a) 뫉/뫉 (b) Percent Approximate Solutions
Fig. 2.1: Simulation results for Theorem 2.1 applied to NN with L2 distance. Left: The ratio between the maximum distance between a query and data points δâ, to the minimum distance δâ. The shaded region shows one stan- dard deviation. As dimensionality increases, this ratio tends to 1. Right: The percentage of data points whose distance to a query is at most (1 + ϵ/100)δâ, visualized for the Gaussian distributionâthe trend is similar for other dis- tributions. As d increases, more vectors fall into the enlarged ball, making them valid solutions to the approximate NN problem. | 2401.09350#83 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 83,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "2.3 Empirical Demonstration of Instability\n21\nG00 0.1) --©-Exp(1) â>-U(0, V2) 10! 10° DIMENSIONALITY (d)\n100 iS} E s0 5 n t 60 B 3 = 40 5 a inf ->-64 & 20 âe128 < $256 3 0 12 0 25 50 75 100 ⬠AS PERCENT 6°\n(a) δâ/δâ (b) Percent Approximate Solutions \nFig. 2.1: Simulation results for Theorem 2.1 applied to NN with L2 distance. Left: The ratio between the maximum distance between a query and data points δâ, to the minimum distance δâ. The shaded region shows one stan- dard deviation. As dimensionality increases, this ratio tends to 1. Right: The percentage of data points whose distance to a query is at most (1 + ϵ/100)δâ, visualized for the Gaussian distributionâthe trend is similar for other dis- tributions. As d increases, more vectors fall into the enlarged ball, making them valid solutions to the approximate NN problem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 84 | that, as d increases, we can find a smaller ϵ such that nearly all data points fall within (1 + ϵ)δâ distance from the query. The results of our experiments confirm this phenomenon; we have plotted the results for the Gaussian dis- tribution in Figure 2.1(b).
# 2.3.1 Maximum Inner Product Search
In the discussion above, we established that retrieval becomes unstable in high dimensions if the data satisfies certain statistical conditions. That meant that the difference between the maximum and the minimum distance grows just as fast as the magnitude of the minimum distance, so that any approximate solution becomes meaningless.
The instability statement does not necessarily imply, however, that the distances become small or converge to a certain value. But as we see in this section, inner product in high dimensions does become smaller and smaller as a function of d.
22
2 Retrieval Stability in High Dimensions
The following theorem summarizes this phenomenon for a unit query point and bounded data points. Note that, the condition that q is a unit vector is not restrictive in any way, as the norm of the query point does not affect the retrieval outcome.
Theorem 2.2 If m data points with bounded norms, and a unit query vector q are drawn iid from a spherically symmetric1 distribution in Rd, then:
jm, P[(a.X) >] =0. | 2401.09350#84 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 84,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "that, as d increases, we can find a smaller ϵ such that nearly all data points fall within (1 + ϵ)δâ distance from the query. The results of our experiments confirm this phenomenon; we have plotted the results for the Gaussian dis- tribution in Figure 2.1(b).\n# 2.3.1 Maximum Inner Product Search\nIn the discussion above, we established that retrieval becomes unstable in high dimensions if the data satisfies certain statistical conditions. That meant that the difference between the maximum and the minimum distance grows just as fast as the magnitude of the minimum distance, so that any approximate solution becomes meaningless.\nThe instability statement does not necessarily imply, however, that the distances become small or converge to a certain value. But as we see in this section, inner product in high dimensions does become smaller and smaller as a function of d.\n22\n2 Retrieval Stability in High Dimensions\nThe following theorem summarizes this phenomenon for a unit query point and bounded data points. Note that, the condition that q is a unit vector is not restrictive in any way, as the norm of the query point does not affect the retrieval outcome.\nTheorem 2.2 If m data points with bounded norms, and a unit query vector q are drawn iid from a spherically symmetric1 distribution in Rd, then:\njm, P[(a.X) >] =0.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 85 | jm, P[(a.X) >] =0.
Proof. By spherical symmetry, it is easy to see that E[â¨q, Xâ©] = 0. The vari- ance of the inner product is then equal to E[â¨q, Xâ©2], which can be expanded as follows.
First, find an orthogonal transformation Î : Rd â Rd that maps the query point q to the first standard basis (i.e., e1 = [1, 0, 0, . . . , 0] â Rd). Due to spherical symmetry, this transformation does not change the data distribution. Now, we can write:
E[(q, X)"] = El('q, PX)"] = E[(PX)q] = i< 1 e(5 rirxya) = Axi. i=1
In the above, the third equality is due to the fact that the distribution of the (transformed) vectors is the same in every direction. Because â¥X⥠is bounded by assumption, the variance of the inner product between q and a random ââ data point tends to 0 as d â â. The claim follows. | 2401.09350#85 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 85,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "jm, P[(a.X) >] =0.\nProof. By spherical symmetry, it is easy to see that E[â¨q, Xâ©] = 0. The vari- ance of the inner product is then equal to E[â¨q, Xâ©2], which can be expanded as follows.\nFirst, find an orthogonal transformation Î : Rd â Rd that maps the query point q to the first standard basis (i.e., e1 = [1, 0, 0, . . . , 0] â Rd). Due to spherical symmetry, this transformation does not change the data distribution. Now, we can write:\nE[(q, X)\"] = El('q, PX)\"] = E[(PX)q] = i< 1 e(5 rirxya) = Axi. i=1\nIn the above, the third equality is due to the fact that the distribution of the (transformed) vectors is the same in every direction. Because â¥X⥠is bounded by assumption, the variance of the inner product between q and a random ââ data point tends to 0 as d â â. The claim follows.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 86 | The proof of Theorem 2.2 tells us that the variance of inner product grows as a function of 1/d and â¥Xâ¥2 2. So if our vectors have bounded norms, then we can find a d such that inner products are arbitrarily close to 0. This is yet another reason that approximate MIPS could become meaningless. But if our data points are clustered in (near) orthogonal subspaces, then approximate MIPS over clusters makes senseâthough, again, MIPS within clusters would be unstable.
# References
K. Beyer, J. Goldstein, R. Ramakrishnan, and U. Shaft. When is ânearest neighborâ meaningful? In Database Theory, pages 217â235, 1999.
1 A distribution is spherically symmetric if it remains invariant under an orthogonal trans- formation.
Chapter 3 Intrinsic Dimensionality | 2401.09350#86 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 86,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The proof of Theorem 2.2 tells us that the variance of inner product grows as a function of 1/d and â¥Xâ¥2 2. So if our vectors have bounded norms, then we can find a d such that inner products are arbitrarily close to 0. This is yet another reason that approximate MIPS could become meaningless. But if our data points are clustered in (near) orthogonal subspaces, then approximate MIPS over clusters makes senseâthough, again, MIPS within clusters would be unstable.\n# References\nK. Beyer, J. Goldstein, R. Ramakrishnan, and U. Shaft. When is ânearest neighborâ meaningful? In Database Theory, pages 217â235, 1999.\n1 A distribution is spherically symmetric if it remains invariant under an orthogonal trans- formation.\nChapter 3 Intrinsic Dimensionality",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 87 | 1 A distribution is spherically symmetric if it remains invariant under an orthogonal trans- formation.
Chapter 3 Intrinsic Dimensionality
Abstract We have seen that high dimensionality poses difficulties for vector retrieval. Yet, judging by the progression from hand-crafted feature vectors to sophisticated embeddings of data, we detect a clear trend towards higher dimensional representations of data. How worried should we be about this ever increasing dimensionality? This chapter explores that question. Its key message is that, even though data points may appear to belong to a high- dimensional space, they actually lie on or near a low-dimensional manifold and, as such, have a low intrinsic dimensionality. This chapter then formalizes the notion of intrinsic dimensionality and presents a mathematical framework that will be useful in analyses in future chapters.
# 3.1 High-Dimensional Data and Low-Dimensional Manifolds | 2401.09350#87 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 87,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "1 A distribution is spherically symmetric if it remains invariant under an orthogonal trans- formation.\nChapter 3 Intrinsic Dimensionality\nAbstract We have seen that high dimensionality poses difficulties for vector retrieval. Yet, judging by the progression from hand-crafted feature vectors to sophisticated embeddings of data, we detect a clear trend towards higher dimensional representations of data. How worried should we be about this ever increasing dimensionality? This chapter explores that question. Its key message is that, even though data points may appear to belong to a high- dimensional space, they actually lie on or near a low-dimensional manifold and, as such, have a low intrinsic dimensionality. This chapter then formalizes the notion of intrinsic dimensionality and presents a mathematical framework that will be useful in analyses in future chapters.\n# 3.1 High-Dimensional Data and Low-Dimensional Manifolds",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 88 | # 3.1 High-Dimensional Data and Low-Dimensional Manifolds
We talked a lot about the difficulties of answering ϵ-approximate top-k ques- tions in high dimensions. We said, in certain situations, the question itself becomes meaningless and retrieval falls apart. For MIPS, in particular, we argued in Theorem 2.2 that points become nearly orthogonal almost surely as the number of dimensions increases. But how concerned should we be, especially given the ever-increasing dimensionality of vector representations of data? Do our data points really live in such extremely high-dimensional spaces? Are all the dimensions necessary to preserving the structure of our data or do our data points have an intrinsically smaller dimensionality?
The answer to these questions is sometimes obvious. If a set of points in Rd lie strictly in a flat subspace Rd⦠with d⦠< d, then one can simply drop the âunusedâ dimensionsâperhaps after a rotation. This could happen if a pair of coordinates are correlated, for instance. No matter what query vector we are performing retrieval for or what distance function we use, the top-k
23
24
3 Intrinsic Dimensionality
set does not change whether the unused dimensions are taken into account or the vectors corrected to lie in Rd⦠. | 2401.09350#88 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 88,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 3.1 High-Dimensional Data and Low-Dimensional Manifolds\nWe talked a lot about the difficulties of answering ϵ-approximate top-k ques- tions in high dimensions. We said, in certain situations, the question itself becomes meaningless and retrieval falls apart. For MIPS, in particular, we argued in Theorem 2.2 that points become nearly orthogonal almost surely as the number of dimensions increases. But how concerned should we be, especially given the ever-increasing dimensionality of vector representations of data? Do our data points really live in such extremely high-dimensional spaces? Are all the dimensions necessary to preserving the structure of our data or do our data points have an intrinsically smaller dimensionality?\nThe answer to these questions is sometimes obvious. If a set of points in Rd lie strictly in a flat subspace Rd⦠with d⦠< d, then one can simply drop the âunusedâ dimensionsâperhaps after a rotation. This could happen if a pair of coordinates are correlated, for instance. No matter what query vector we are performing retrieval for or what distance function we use, the top-k\n23\n24\n3 Intrinsic Dimensionality\nset does not change whether the unused dimensions are taken into account or the vectors corrected to lie in Rd⦠.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 89 | 23
24
3 Intrinsic Dimensionality
set does not change whether the unused dimensions are taken into account or the vectors corrected to lie in Rd⦠.
Other times the answer is intuitive but not so obvious. When a text doc- ument is represented as a sparse vector, all the documentâs information is contained entirely in the vectorâs non-zero coordinates. The coordinates that are 0 do not contribute to the representation of the document in any way. In a sense then, the intrinsic dimensionality of a collection of such sparse vectors is in the order of the number of non-zero coordinates, rather than the nominal dimensionality of the space the points lie in.
It appears then that there are instances where a collection of points have a superficially large number of dimensions, d, but that, in fact, the points lie in a lower-dimensional space with dimensionality dâ¦. We call d⦠the intrinsic dimensionality of the point set.
This situation, where the intrinsic dimensionality of data is lower than that of the space, arises more commonly than one imagines. In fact, so common is this phenomenon that in statistical learning theory, there are special classes of algorithms [Ma and Fu, 2012] designed for data collections that lie on or near a low-dimensional submanifold of Rd despite their apparent arbitrarily high-dimensional representations. | 2401.09350#89 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 89,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "23\n24\n3 Intrinsic Dimensionality\nset does not change whether the unused dimensions are taken into account or the vectors corrected to lie in Rd⦠.\nOther times the answer is intuitive but not so obvious. When a text doc- ument is represented as a sparse vector, all the documentâs information is contained entirely in the vectorâs non-zero coordinates. The coordinates that are 0 do not contribute to the representation of the document in any way. In a sense then, the intrinsic dimensionality of a collection of such sparse vectors is in the order of the number of non-zero coordinates, rather than the nominal dimensionality of the space the points lie in.\nIt appears then that there are instances where a collection of points have a superficially large number of dimensions, d, but that, in fact, the points lie in a lower-dimensional space with dimensionality dâ¦. We call d⦠the intrinsic dimensionality of the point set.\nThis situation, where the intrinsic dimensionality of data is lower than that of the space, arises more commonly than one imagines. In fact, so common is this phenomenon that in statistical learning theory, there are special classes of algorithms [Ma and Fu, 2012] designed for data collections that lie on or near a low-dimensional submanifold of Rd despite their apparent arbitrarily high-dimensional representations.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 90 | In the context of vector retrieval, too, the concept of intrinsic dimensional- ity often plays an important role. Knowing that data points have a low intrin- sic dimensionality means we may be able to reduce dimensionality without (substantially) losing the geometric structure of the data, including inter- point distances. But more importantly, we can design algorithms specifically for data with low intrinsic dimensionality, as we will see in later chapters. In our analysis of many of these algorithms, too, we often resort to this property to derive meaningful bounds and make assertions about their performance. Doing so, however, requires that we formalize the notion of intrinsic dimen- sionality. We often do not have a characterization of the submanifold itself, so we need an alternate way of characterizing the low-dimensional structure of our data points. In the remainder of this chapter, we present two com- mon (and related) definitions of intrinsic dimensionality that will be useful in subsequent chapters.
# 3.2 Doubling Measure and Expansion Rate
Karger and Ruhl [2002] characterize intrinsic dimensionality as the growth or expansion rate of a point set. To understand what that means intuitively, place yourself somewhere in the data collection, draw a ball around yourself, and count how many data points are in that ball. Now expand the radius of
3.2 Doubling Measure and Expansion Rate | 2401.09350#90 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 90,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In the context of vector retrieval, too, the concept of intrinsic dimensional- ity often plays an important role. Knowing that data points have a low intrin- sic dimensionality means we may be able to reduce dimensionality without (substantially) losing the geometric structure of the data, including inter- point distances. But more importantly, we can design algorithms specifically for data with low intrinsic dimensionality, as we will see in later chapters. In our analysis of many of these algorithms, too, we often resort to this property to derive meaningful bounds and make assertions about their performance. Doing so, however, requires that we formalize the notion of intrinsic dimen- sionality. We often do not have a characterization of the submanifold itself, so we need an alternate way of characterizing the low-dimensional structure of our data points. In the remainder of this chapter, we present two com- mon (and related) definitions of intrinsic dimensionality that will be useful in subsequent chapters.\n# 3.2 Doubling Measure and Expansion Rate\nKarger and Ruhl [2002] characterize intrinsic dimensionality as the growth or expansion rate of a point set. To understand what that means intuitively, place yourself somewhere in the data collection, draw a ball around yourself, and count how many data points are in that ball. Now expand the radius of\n3.2 Doubling Measure and Expansion Rate",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 91 | 3.2 Doubling Measure and Expansion Rate
this ball by a factor 2, and count again. The count of data points in a âgrowth- restrictedâ point set should increase smoothly, rather than suddenly, as we make this ball larger.
In other words, data points âcome into view,â as Karger and Ruhl [2002] put it, at a constant rate as we expand our view, regardless of where we are located. We will not encounter massive holes in the space where there are no data points, followed abruptly by a region where a large number of vectors are concentrated.
The formal definition is not far from the intuitive description above. In fact, expansion rate as defined by Karger and Ruhl [2002] is an instance of the following more general definition of a doubling measure, where the measure µ is the counting measure over a collection of points X .
Definition 3.1 A distribution µ on Rd is a doubling measure if there is a constant d⦠such that, for any r > 0 and x â Rd, µ(B(x, 2r)) ⤠2d⦠µ(B(x, r)). The constant d⦠is said to be the expansion rate of the distribution. | 2401.09350#91 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 91,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "3.2 Doubling Measure and Expansion Rate\nthis ball by a factor 2, and count again. The count of data points in a âgrowth- restrictedâ point set should increase smoothly, rather than suddenly, as we make this ball larger.\nIn other words, data points âcome into view,â as Karger and Ruhl [2002] put it, at a constant rate as we expand our view, regardless of where we are located. We will not encounter massive holes in the space where there are no data points, followed abruptly by a region where a large number of vectors are concentrated.\nThe formal definition is not far from the intuitive description above. In fact, expansion rate as defined by Karger and Ruhl [2002] is an instance of the following more general definition of a doubling measure, where the measure µ is the counting measure over a collection of points X .\nDefinition 3.1 A distribution µ on Rd is a doubling measure if there is a constant d⦠such that, for any r > 0 and x â Rd, µ(B(x, 2r)) ⤠2d⦠µ(B(x, r)). The constant d⦠is said to be the expansion rate of the distribution.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 92 | One can think of the expansion rate d⦠as a dimension of sorts. In fact, as we will see later, several works [Dasgupta and Sinha, 2015, Karger and Ruhl, 2002, Beygelzimer et al., 2006] use this notion of intrinsic dimensionality to design algorithms for top-k retrieval or utilize it to derive performance guarantees for vector collections that are drawn from a doubling measure. That is the main reason we review this definition of intrinsic dimensionality in this chapter.
While the expansion rate is a reasonable way of describing the structure of a set of points, it is unfortunately not a stable indicator. It can suddenly blow up, for example, by the addition of a single point to the set. As a concrete example, consider the set of integers between |r| and |2r| for any arbitrary value of r: X = {u â Z | r < |u| < 2r}. The expansion rate of the resulting set is constant because no matter which point we choose as the center of our ball, and regardless of our choice of radius, doubling the radius brings points into view at a constant rate. | 2401.09350#92 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 92,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "One can think of the expansion rate d⦠as a dimension of sorts. In fact, as we will see later, several works [Dasgupta and Sinha, 2015, Karger and Ruhl, 2002, Beygelzimer et al., 2006] use this notion of intrinsic dimensionality to design algorithms for top-k retrieval or utilize it to derive performance guarantees for vector collections that are drawn from a doubling measure. That is the main reason we review this definition of intrinsic dimensionality in this chapter.\nWhile the expansion rate is a reasonable way of describing the structure of a set of points, it is unfortunately not a stable indicator. It can suddenly blow up, for example, by the addition of a single point to the set. As a concrete example, consider the set of integers between |r| and |2r| for any arbitrary value of r: X = {u â Z | r < |u| < 2r}. The expansion rate of the resulting set is constant because no matter which point we choose as the center of our ball, and regardless of our choice of radius, doubling the radius brings points into view at a constant rate.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 93 | What happens if we added the origin to the set, so that our set becomes {0} ⪠X ? If we chose 0 as the center of the ball, and set its radius to r, we have a single point in the resulting ball. The moment we double r, the resulting ball will contain the entire set! In other words, the expansion rate of the updated set is log m (where m = |X |).
It is easy to argue that a subset of a set with bounded expansion rate does not necessarily have a bounded expansion rate itself. This unstable behavior is less than ideal, which is why a more robust notion of intrinsic dimensionality has been developed. We will introduce that next.
25
26
3 Intrinsic Dimensionality
# 3.3 Doubling Dimension
Another idea to formalize intrinsic dimensionality that has worked well in algorithmic design and anlysis is the doubling dimension. It was introduced by Gupta et al. [2003] but is closely related to the Assouad dimension [As- souad, 1983]. It is defined as follows.
Definition 3.2 A set X â Rd is said to have doubling dimension d⦠if B(·, 2r) â© X , the intersection of any ball of radius 2r with the set, can be covered by at most 2d⦠balls of radius r. | 2401.09350#93 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 93,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "What happens if we added the origin to the set, so that our set becomes {0} ⪠X ? If we chose 0 as the center of the ball, and set its radius to r, we have a single point in the resulting ball. The moment we double r, the resulting ball will contain the entire set! In other words, the expansion rate of the updated set is log m (where m = |X |).\nIt is easy to argue that a subset of a set with bounded expansion rate does not necessarily have a bounded expansion rate itself. This unstable behavior is less than ideal, which is why a more robust notion of intrinsic dimensionality has been developed. We will introduce that next.\n25\n26\n3 Intrinsic Dimensionality\n# 3.3 Doubling Dimension\nAnother idea to formalize intrinsic dimensionality that has worked well in algorithmic design and anlysis is the doubling dimension. It was introduced by Gupta et al. [2003] but is closely related to the Assouad dimension [As- souad, 1983]. It is defined as follows.\nDefinition 3.2 A set X â Rd is said to have doubling dimension d⦠if B(·, 2r) â© X , the intersection of any ball of radius 2r with the set, can be covered by at most 2d⦠balls of radius r.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 94 | The base 2 in the definition above can be replaced with any other constant k: The doubling dimension of X is d⦠if the intersection of any ball of radius r with the set can be covered by O(kd⦠) balls of radius r/k. Furthermore, the definition can be easily extended to any metric space, not just Rd with the Euclidean norm.
The doubling dimension is a different notion from the expansion rate as defined in Definition 3.1. The two, however, are in some sense related, as the following lemma shows.
Lemma 3.1 The doubling dimension, d⦠of any finite metric (X, δ) is bounded above by its expansion rate, dkr â¦
Proof. Fix a ball B(u, 2r) and let S' be its r-net. That is, S C X, the distance between any two points in S$ is at least r, and ¥ C U,,c5 B(u,r). We have hat:
B(u, 2r) â B(v, r) â B(u, 4r). vâS
By definition of the expansion rate, for every v â S:
|B(u, 4r)| < |B(v, 8r)| < gadct Biv, 5). | 2401.09350#94 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 94,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The base 2 in the definition above can be replaced with any other constant k: The doubling dimension of X is d⦠if the intersection of any ball of radius r with the set can be covered by O(kd⦠) balls of radius r/k. Furthermore, the definition can be easily extended to any metric space, not just Rd with the Euclidean norm.\nThe doubling dimension is a different notion from the expansion rate as defined in Definition 3.1. The two, however, are in some sense related, as the following lemma shows.\nLemma 3.1 The doubling dimension, d⦠of any finite metric (X, δ) is bounded above by its expansion rate, dkr â¦\nProof. Fix a ball B(u, 2r) and let S' be its r-net. That is, S C X, the distance between any two points in S$ is at least r, and Â¥ C U,,c5 B(u,r). We have hat:\nB(u, 2r) â B(v, r) â B(u, 4r). vâS\nBy definition of the expansion rate, for every v â S:\n|B(u, 4r)| < |B(v, 8r)| < gadct Biv, 5).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 95 | By definition of the expansion rate, for every v â S:
|B(u, 4r)| < |B(v, 8r)| < gadct Biv, 5).
Because the balls B(v, r/2) for all v â S are disjoint, it follows that |S| ⤠24dkr ⦠many balls of radius r cover B(u, 2r). That concludes the ââ proof.
The doubling dimension and expansion rate both quantify the intrinsic dimensionality of a point set. But Lemma 3.1 shows that, the class of doubling metrics (i.e., metric spaces with a constant doubling dimen- sion) contains the class of metrics with a bounded expansion rate.
The converse of the above lemma is not true. In other words, there are sets that have a bounded doubling dimension, but whose expansion rate is unbounded. The set, X = {0} ⪠{u â Z | r < |u| < 2r}, from the previous
3.3 Doubling Dimension
section is one example where this happens. From our discussion above, its expansion rate is log|X |. It is easy to see that the doubling dimension of this set, however, is constant.
# 3.3.1 Properties of the Doubling Dimension | 2401.09350#95 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 95,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "By definition of the expansion rate, for every v â S:\n|B(u, 4r)| < |B(v, 8r)| < gadct Biv, 5).\nBecause the balls B(v, r/2) for all v â S are disjoint, it follows that |S| ⤠24dkr ⦠many balls of radius r cover B(u, 2r). That concludes the ââ proof.\nThe doubling dimension and expansion rate both quantify the intrinsic dimensionality of a point set. But Lemma 3.1 shows that, the class of doubling metrics (i.e., metric spaces with a constant doubling dimen- sion) contains the class of metrics with a bounded expansion rate.\nThe converse of the above lemma is not true. In other words, there are sets that have a bounded doubling dimension, but whose expansion rate is unbounded. The set, X = {0} ⪠{u â Z | r < |u| < 2r}, from the previous\n3.3 Doubling Dimension\nsection is one example where this happens. From our discussion above, its expansion rate is log|X |. It is easy to see that the doubling dimension of this set, however, is constant.\n# 3.3.1 Properties of the Doubling Dimension",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 96 | # 3.3.1 Properties of the Doubling Dimension
It is helpful to go over a few concrete examples of point sets with bounded doubling dimension in order to understand a few properties of this definition of intrinsic dimensionality. We will start with a simple example: a line segment in Rd with the Euclidean norm.
If the set X is a line segment, then its intersection with a ball of radius r is itself a line segment. Clearly, the intersection set can be covered with two balls of radius r/2. Therefore, the doubling dimension d⦠of X is 1.
We can extend that result to any affine set in Rd to obtain the following
property:
Lemma 3.2 A k-dimensional flat in Rd has doubling dimension O(k).
Proof. The intersection of a ball in Rd and a k-dimensional flat is a ball in Rk. It is a well-known result that the size of an ϵ-net of a unit ball in Rk is at most (C/ϵ)k for some small constant C. As such, a ball of radius r can be covered with 2O(k) balls of radius r/2, implying the claim. ââ | 2401.09350#96 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 96,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 3.3.1 Properties of the Doubling Dimension\nIt is helpful to go over a few concrete examples of point sets with bounded doubling dimension in order to understand a few properties of this definition of intrinsic dimensionality. We will start with a simple example: a line segment in Rd with the Euclidean norm.\nIf the set X is a line segment, then its intersection with a ball of radius r is itself a line segment. Clearly, the intersection set can be covered with two balls of radius r/2. Therefore, the doubling dimension d⦠of X is 1.\nWe can extend that result to any affine set in Rd to obtain the following\nproperty:\nLemma 3.2 A k-dimensional flat in Rd has doubling dimension O(k).\nProof. The intersection of a ball in Rd and a k-dimensional flat is a ball in Rk. It is a well-known result that the size of an ϵ-net of a unit ball in Rk is at most (C/ϵ)k for some small constant C. As such, a ball of radius r can be covered with 2O(k) balls of radius r/2, implying the claim. ââ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 97 | The lemma above tells us that the doubling dimension of a set in the Euclidean space is at most some constant factor larger than the natural di- mension of the space; note that this was not the case for the expansion rate. Another important property that speaks to the stability of the doubling di- mension is the following, which is trivially true:
Lemma 3.3 Any subset of a set with doubling dimension d⦠itself has dou- bling dimension dâ¦.
The doubling dimension is also robust under the addition of points to the set, as the following result shows.
Lemma 3.4 Suppose sets Xi for i â [n] each have doubling dimension dâ¦. Then their union has doubling dimension at most d⦠+ log n.
Proof. For any ball B of radius r, B â© Xi can be covered with 2d⦠balls of half the radius. As such, at most n2d⦠balls of radius r/2 are needed to cover ââ the union. The doubling dimension of the union is therefore d⦠+ log n.
One consequence of the previous two lemmas is the following statement concerning sparse vectors:
27
28
3 Intrinsic Dimensionality | 2401.09350#97 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 97,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The lemma above tells us that the doubling dimension of a set in the Euclidean space is at most some constant factor larger than the natural di- mension of the space; note that this was not the case for the expansion rate. Another important property that speaks to the stability of the doubling di- mension is the following, which is trivially true:\nLemma 3.3 Any subset of a set with doubling dimension d⦠itself has dou- bling dimension dâ¦.\nThe doubling dimension is also robust under the addition of points to the set, as the following result shows.\nLemma 3.4 Suppose sets Xi for i â [n] each have doubling dimension dâ¦. Then their union has doubling dimension at most d⦠+ log n.\nProof. For any ball B of radius r, B â© Xi can be covered with 2d⦠balls of half the radius. As such, at most n2d⦠balls of radius r/2 are needed to cover ââ the union. The doubling dimension of the union is therefore d⦠+ log n.\nOne consequence of the previous two lemmas is the following statement concerning sparse vectors:\n27\n28\n3 Intrinsic Dimensionality",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 98 | One consequence of the previous two lemmas is the following statement concerning sparse vectors:
27
28
3 Intrinsic Dimensionality
Lemma 3.5 Suppose that X C R¢@ is a collection of sparse vectors, each having at most n non-zero coordinates. Then the doubling dimension of X is at most Ck + klogd for some constant C. Proof. X is the union of (2) < dâ n-dimensional flats. Each of these flats has doubling dimension Ck for some universal constant Câ, by Lemma 3.2. By the application of Lemma 3.4, we get that the doubling dimension of Â¥ is at most Cn + nlog d. Qo
Lemma 3.5 states that collections of sparse vectors in the Euclidean space are naturally described by the doubling dimension.
# References
P. Assouad. Plongements lipschitziens dans Rn. Bulletin de la Soci´et´e
Math´ematique de France, 111:429â448, 1983.
A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. In Proceedings of the 23rd International Conference on Machine Learning, page 97â104, 2006.
S. Dasgupta and K. Sinha. Randomized partition trees for nearest neighbor search. Algorithmica, 72(1):237â263, 5 2015. | 2401.09350#98 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 98,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "One consequence of the previous two lemmas is the following statement concerning sparse vectors:\n27\n28\n3 Intrinsic Dimensionality\nLemma 3.5 Suppose that X C R¢@ is a collection of sparse vectors, each having at most n non-zero coordinates. Then the doubling dimension of X is at most Ck + klogd for some constant C. Proof. X is the union of (2) < dâ n-dimensional flats. Each of these flats has doubling dimension Ck for some universal constant Câ, by Lemma 3.2. By the application of Lemma 3.4, we get that the doubling dimension of Â¥ is at most Cn + nlog d. Qo\nLemma 3.5 states that collections of sparse vectors in the Euclidean space are naturally described by the doubling dimension.\n# References\nP. Assouad. Plongements lipschitziens dans Rn. Bulletin de la Soci´et´e\nMath´ematique de France, 111:429â448, 1983.\nA. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. In Proceedings of the 23rd International Conference on Machine Learning, page 97â104, 2006.\nS. Dasgupta and K. Sinha. Randomized partition trees for nearest neighbor search. Algorithmica, 72(1):237â263, 5 2015.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 99 | S. Dasgupta and K. Sinha. Randomized partition trees for nearest neighbor search. Algorithmica, 72(1):237â263, 5 2015.
A. Gupta, R. Krauthgamer, and J. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Annual IEEE Symposium on Founda- tions of Computer Science, pages 534â543, 2003.
D. R. Karger and M. Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 741â750, 2002.
Y. Ma and Y. Fu. Manifold Learning Theory and Applications. CRC Press, 2012.
Part II Retrieval Algorithms
# Chapter 4 Branch-and-Bound Algorithms
Abstract One of the earliest approaches to the top-k retrieval problem is to partition the vector space recursively into smaller regions and, each time we do so, make note of their geometry. During search, we eliminate the regions whose shape indicates they cannot contain or overlap with the solution set. This chapter covers algorithms that embody this approach and discusses their exact and approximate variants.
# 4.1 Intuition | 2401.09350#99 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 99,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Dasgupta and K. Sinha. Randomized partition trees for nearest neighbor search. Algorithmica, 72(1):237â263, 5 2015.\nA. Gupta, R. Krauthgamer, and J. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Annual IEEE Symposium on Founda- tions of Computer Science, pages 534â543, 2003.\nD. R. Karger and M. Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 741â750, 2002.\nY. Ma and Y. Fu. Manifold Learning Theory and Applications. CRC Press, 2012.\nPart II Retrieval Algorithms\n# Chapter 4 Branch-and-Bound Algorithms\nAbstract One of the earliest approaches to the top-k retrieval problem is to partition the vector space recursively into smaller regions and, each time we do so, make note of their geometry. During search, we eliminate the regions whose shape indicates they cannot contain or overlap with the solution set. This chapter covers algorithms that embody this approach and discusses their exact and approximate variants.\n# 4.1 Intuition",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 100 | # 4.1 Intuition
Suppose there was some way to split a collection X into two sub-collections, Xl and Xr, such that X = Xl ⪠Xr and that the two sub-collections have roughly the same size. In general, we can relax the splitting criterion so the two sub-collections are not necessarily partitions; that is, we may have Xl â© Xr ̸= â
. We may also split the collection into more than two sub-collections. For the moment, though, assume we have two sub-collections that do not overlap.
Suppose further that, we could geometrically characterize exactly the re- gions that contain Xl and Xr. For example, when Xl â© Xr = â
, these regions partition the space and may therefore be characterized by a separating hyper- plane. Call these regions Rl and Rr, respectively. The separating hyperplane forms a decision boundary that helps us determine if a vector falls into Rl or Rr.
In effect, we have created a binary tree of depth 1 where the root node has a decision boundary and each of the two leaves contains data points that fall into its region. This is illustrated in Figure 4.1(a). | 2401.09350#100 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 100,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 4.1 Intuition\nSuppose there was some way to split a collection X into two sub-collections, Xl and Xr, such that X = Xl ⪠Xr and that the two sub-collections have roughly the same size. In general, we can relax the splitting criterion so the two sub-collections are not necessarily partitions; that is, we may have Xl â© Xr ̸= â
. We may also split the collection into more than two sub-collections. For the moment, though, assume we have two sub-collections that do not overlap.\nSuppose further that, we could geometrically characterize exactly the re- gions that contain Xl and Xr. For example, when Xl â© Xr = â
, these regions partition the space and may therefore be characterized by a separating hyper- plane. Call these regions Rl and Rr, respectively. The separating hyperplane forms a decision boundary that helps us determine if a vector falls into Rl or Rr.\nIn effect, we have created a binary tree of depth 1 where the root node has a decision boundary and each of the two leaves contains data points that fall into its region. This is illustrated in Figure 4.1(a).",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 102 | Fig. 4.1: Illustration of a general branch-and-bound method on a toy collec- tion in R2. In (a), Rl and Rr are separated by the dashed line h. The distance between query q and the closest vector in Rl is less than the distance be- tween q and h. As such, we do not need to search for the top-1 vector over the points in Rr, so that the right branch of the tree is pruned. In (b), the regions are recursively split until each terminal region contains at most two data points. We then find the distance between q and the data points in the region that contains q, G. If the ball around q with this distance as its radius does not intersect a region, we can safely prune that regionâregions that are not shaded in the figure. Otherwise, we may have to search it during the certification process.
and navigating to the appropriate leaf. Now, we solve the exact top-1 retrieval problem over Xl to obtain the optimal point in that region uâ l , then make a note of this minimum distance obtained, δ(q, uâ At this point, if it turns out that δ(q, uâ | 2401.09350#102 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 102,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 4.1: Illustration of a general branch-and-bound method on a toy collec- tion in R2. In (a), Rl and Rr are separated by the dashed line h. The distance between query q and the closest vector in Rl is less than the distance be- tween q and h. As such, we do not need to search for the top-1 vector over the points in Rr, so that the right branch of the tree is pruned. In (b), the regions are recursively split until each terminal region contains at most two data points. We then find the distance between q and the data points in the region that contains q, G. If the ball around q with this distance as its radius does not intersect a region, we can safely prune that regionâregions that are not shaded in the figure. Otherwise, we may have to search it during the certification process.\nand navigating to the appropriate leaf. Now, we solve the exact top-1 retrieval problem over Xl to obtain the optimal point in that region uâ l , then make a note of this minimum distance obtained, δ(q, uâ At this point, if it turns out that δ(q, uâ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 103 | l ) < δ(q, Rr)1 then we have found the optimal point and do not need to search the data points in Xr at all! That is because, the δ-ball2 centered at q with radius δ(q, uâ l ) is contained entirely in Rl, so that no point from Rr can have a shorter distance to q than uâ l . Refer again to Figure 4.1(a) for an illustration of this scenario.
l ) ⥠δ(q, Rr), then we proceed to solve the top-1 problem over Xr as well and compare the solution with uâ l to find the optimal vector. We can think of the comparison between δ(q, uâ l ) with δ(q, Rr) as backtracking to the parent node of Rl in the equivalent treeâ which is the rootâand comparing δ(q, uâ l ) with the distance of q with the decision boundary. This process of backtracking and deciding to prune a
1 The distance between a point u and a set S is defined as δ(u, S) = inf vâS δ(u, v). 2 The ball centered at u with radius r with respect to metric δ is {x | δ(u, x) ⤠r}.
4.2 k-dimensional Trees | 2401.09350#103 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 103,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "l ) < δ(q, Rr)1 then we have found the optimal point and do not need to search the data points in Xr at all! That is because, the δ-ball2 centered at q with radius δ(q, uâ l ) is contained entirely in Rl, so that no point from Rr can have a shorter distance to q than uâ l . Refer again to Figure 4.1(a) for an illustration of this scenario.\nl ) ⥠δ(q, Rr), then we proceed to solve the top-1 problem over Xr as well and compare the solution with uâ l to find the optimal vector. We can think of the comparison between δ(q, uâ l ) with δ(q, Rr) as backtracking to the parent node of Rl in the equivalent treeâ which is the rootâand comparing δ(q, uâ l ) with the distance of q with the decision boundary. This process of backtracking and deciding to prune a\n1 The distance between a point u and a set S is defined as δ(u, S) = inf vâS δ(u, v). 2 The ball centered at u with radius r with respect to metric δ is {x | δ(u, x) ⤠r}.\n4.2 k-dimensional Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 104 | 4.2 k-dimensional Trees
branch or search it certifies that uâ l top-1 problem exactly. is indeed optimal, thereby solving the
We can extend the framework above easily by recursively splitting the two sub-collections and characterizing the regions containing the resulting partitions. This leads to a (balanced) binary tree where each internal node has a decision boundaryâthe separating hyperplane of its child regions. We may stop splitting a node if it has fewer than m⦠points. This extension is rendered in Figure 4.1(b).
The retrieval process is the same but needs a little more care: Let the query q traverse the tree from root to leaf, where each internal node determines if q belongs to the âleftâ or ârightâ sub-regions and routes q accordingly. Once we have found the leaf (terminal) region that contains q, we find the candidate vector uâ, then backtrack and certify that uâ is indeed optimal.
During the backtracking, at each internal node, we compare the distance between q and the current candidate with the distance between q and the region on the other side of the decision boundary. As before, that comparison results in either pruning a branch or searching it to find a possibly better candidate. The certification process stops when we find ourselves back in the root node with no more branches to verify, at which point we have found the optimal solution. | 2401.09350#104 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 104,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "4.2 k-dimensional Trees\nbranch or search it certifies that uâ l top-1 problem exactly. is indeed optimal, thereby solving the\nWe can extend the framework above easily by recursively splitting the two sub-collections and characterizing the regions containing the resulting partitions. This leads to a (balanced) binary tree where each internal node has a decision boundaryâthe separating hyperplane of its child regions. We may stop splitting a node if it has fewer than m⦠points. This extension is rendered in Figure 4.1(b).\nThe retrieval process is the same but needs a little more care: Let the query q traverse the tree from root to leaf, where each internal node determines if q belongs to the âleftâ or ârightâ sub-regions and routes q accordingly. Once we have found the leaf (terminal) region that contains q, we find the candidate vector uâ, then backtrack and certify that uâ is indeed optimal.\nDuring the backtracking, at each internal node, we compare the distance between q and the current candidate with the distance between q and the region on the other side of the decision boundary. As before, that comparison results in either pruning a branch or searching it to find a possibly better candidate. The certification process stops when we find ourselves back in the root node with no more branches to verify, at which point we have found the optimal solution.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 105 | The above is the logic that is at the core of branch-and-bound algorithms for top-k retrieval [Dasgupta and Sinha, 2015, Bentley, 1975, Ram and Sinha, 2019, Ciaccia et al., 1997, Yianilos, 1993, Liu et al., 2004, Panigrahy, 2008, Ram and Gray, 2012, Bachrach et al., 2014]. The specific instances of this framework differ in terms of how they split a collection and the details of the certification process. We will review key algorithms that belong to this family in the remainder of this chapter. We emphasize that, most branch-and- bound algorithms only address the NN problem in the Euclidean space (so that δ(u, v) = â¥u â vâ¥2) or in growth-restricted measures [Karger and Ruhl, 2002, Clarkson, 1997, Krauthgamer and Lee, 2004] but where the metric is nonetheless proper.
# 4.2 k-dimensional Trees
The k-dimensional Tree or k-d Tree [Bentley, 1975] is a special instance of the framework described above wherein the distance function is Euclidean and the space is recursively partitioned into hyper-rectangles. In other words, the decision boundaries in a k-d Tree are axis-aligned hyperplanes. | 2401.09350#105 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 105,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The above is the logic that is at the core of branch-and-bound algorithms for top-k retrieval [Dasgupta and Sinha, 2015, Bentley, 1975, Ram and Sinha, 2019, Ciaccia et al., 1997, Yianilos, 1993, Liu et al., 2004, Panigrahy, 2008, Ram and Gray, 2012, Bachrach et al., 2014]. The specific instances of this framework differ in terms of how they split a collection and the details of the certification process. We will review key algorithms that belong to this family in the remainder of this chapter. We emphasize that, most branch-and- bound algorithms only address the NN problem in the Euclidean space (so that δ(u, v) = â¥u â vâ¥2) or in growth-restricted measures [Karger and Ruhl, 2002, Clarkson, 1997, Krauthgamer and Lee, 2004] but where the metric is nonetheless proper.\n# 4.2 k-dimensional Trees\nThe k-dimensional Tree or k-d Tree [Bentley, 1975] is a special instance of the framework described above wherein the distance function is Euclidean and the space is recursively partitioned into hyper-rectangles. In other words, the decision boundaries in a k-d Tree are axis-aligned hyperplanes.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 106 | Let us consider its simplest construction for X â Rd. The root of the tree is a node that represents the entire space, which naturally contains the entire data collection. Assuming that the size of the collection is greater than 1, we follow a simple procedure to split the node: We select one coordinate axis and partition the collection at the median of data points along the chosen
33
34
4 Branch-and-Bound Algorithms
direction. The process recurses on each newly-minted node, with nodes at the same depth in the tree using the same coordinate axis for splitting, and where we go through the coordinates in a round-robin manner as the tree grows. We stop splitting a node further if it contains a single data point (m⦠= 1), then mark it as a leaf node.
A few observations that are worth noting. By choosing the median point to split on, we guarantee that the tree is balanced. That together with the fact that m⦠= 1 implies that the depth of the tree is log m where m = |X |. Finally, because nodes in each level of the tree split on the same coordinate, every coordinate is split in (log m)/d levels. These will become important in our analysis of the algorithm.
# 4.2.1 Complexity Analysis | 2401.09350#106 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 106,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Let us consider its simplest construction for X â Rd. The root of the tree is a node that represents the entire space, which naturally contains the entire data collection. Assuming that the size of the collection is greater than 1, we follow a simple procedure to split the node: We select one coordinate axis and partition the collection at the median of data points along the chosen\n33\n34\n4 Branch-and-Bound Algorithms\ndirection. The process recurses on each newly-minted node, with nodes at the same depth in the tree using the same coordinate axis for splitting, and where we go through the coordinates in a round-robin manner as the tree grows. We stop splitting a node further if it contains a single data point (m⦠= 1), then mark it as a leaf node.\nA few observations that are worth noting. By choosing the median point to split on, we guarantee that the tree is balanced. That together with the fact that m⦠= 1 implies that the depth of the tree is log m where m = |X |. Finally, because nodes in each level of the tree split on the same coordinate, every coordinate is split in (log m)/d levels. These will become important in our analysis of the algorithm.\n# 4.2.1 Complexity Analysis",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 107 | # 4.2.1 Complexity Analysis
The k-d Tree data structure is fairly simple to construct. It is also efficient: Its space complexity given a set of m vectors is Î(m) and its construction time has complexity Î(m log m).3
The search algorithm, however, is not so easy to analyze in general. Fried- man et al. [1977] claimed that the expected search complexity is O(log m) for m data points that are sampled uniformly from the unit hypercube. While uniformity is an unrealistic assumption, it is necessary for the analysis of the average case. On the other hand, no generality is lost by the assumption that vectors are contained in the hypercube. That is because, we can always scale every data point by a constant factor into the unit hypercubeâa transfor- mation that does not affect the pairwise distances between vectors. Let us now discuss the sketch of the proof of their claim. | 2401.09350#107 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 107,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 4.2.1 Complexity Analysis\nThe k-d Tree data structure is fairly simple to construct. It is also efficient: Its space complexity given a set of m vectors is Î(m) and its construction time has complexity Î(m log m).3\nThe search algorithm, however, is not so easy to analyze in general. Fried- man et al. [1977] claimed that the expected search complexity is O(log m) for m data points that are sampled uniformly from the unit hypercube. While uniformity is an unrealistic assumption, it is necessary for the analysis of the average case. On the other hand, no generality is lost by the assumption that vectors are contained in the hypercube. That is because, we can always scale every data point by a constant factor into the unit hypercubeâa transfor- mation that does not affect the pairwise distances between vectors. Let us now discuss the sketch of the proof of their claim.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 108 | Let δâ = minuâX â¥q â uâ¥2 be the optimal distance to a query q. Consider the ball of radius δâ centered at q and denote it by B(q, δâ). It is easy to see that the number of leaves we may need to visit in order to certify an initial candidate is upper-bounded by the number of leaf regions (i.e., d-dimensional boxes) that touch B(q, δâ). That quantity itself is upper-bounded by the number of boxes that touch the smallest hypercube that contains B(q, δâ). If we calculated this number, then we have found an upper-bound on the search complexity.
Following the argument above, Friedman et al. [1977] show thatâwith very specific assumptions on the density of vectors in the space, which do not necessarily hold in high dimensionsâthe quantity of interest is upper- bounded by the following expression:
(1+a@'/4)", (4.1)
3 The time to construct the tree depends on the complexity of the subroutine that finds the median of a set of values.
4.2 k-dimensional Trees | 2401.09350#108 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 108,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Let δâ = minuâX â¥q â uâ¥2 be the optimal distance to a query q. Consider the ball of radius δâ centered at q and denote it by B(q, δâ). It is easy to see that the number of leaves we may need to visit in order to certify an initial candidate is upper-bounded by the number of leaf regions (i.e., d-dimensional boxes) that touch B(q, δâ). That quantity itself is upper-bounded by the number of boxes that touch the smallest hypercube that contains B(q, δâ). If we calculated this number, then we have found an upper-bound on the search complexity.\nFollowing the argument above, Friedman et al. [1977] show thatâwith very specific assumptions on the density of vectors in the space, which do not necessarily hold in high dimensionsâthe quantity of interest is upper- bounded by the following expression:\n(1+a@'/4)\", (4.1)\n3 The time to construct the tree depends on the complexity of the subroutine that finds the median of a set of values.\n4.2 k-dimensional Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 109 | 3 The time to construct the tree depends on the complexity of the subroutine that finds the median of a set of values.
4.2 k-dimensional Trees
where G(d) is the ratio between the volume of the hypercube that contains B(q, 뫉) and the volume of B(q, 뫉) itself. Because G(d) is independent of m, and because visiting each leaf takes O(log m) (i.e., the depth of the tree) operations, they conclude that the complexity of the algorithm is O(log m).
# 4.2.2 Failure in High Dimensions
The argument above regarding the search time complexity of the algorithm fails in high dimensions. Let us elaborate why in this section.
Let us accept that the assumptions that enabled the proof above hold and focus on G(d). The volume of a hypercube in d dimensions with sides that have length 2δâ is (2δâ)d. The volume of B(q, δâ) is Ïd/2δâd/Î (d/2+1), where Î denotes the Gamma function. For convenience, suppose that d is even, so that Î (d/2 + 1) = (d/2)!. As such G(d), the ratio between the two volumes, is: | 2401.09350#109 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 109,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "3 The time to construct the tree depends on the complexity of the subroutine that finds the median of a set of values.\n4.2 k-dimensional Trees\nwhere G(d) is the ratio between the volume of the hypercube that contains B(q, δâ) and the volume of B(q, δâ) itself. Because G(d) is independent of m, and because visiting each leaf takes O(log m) (i.e., the depth of the tree) operations, they conclude that the complexity of the algorithm is O(log m).\n# 4.2.2 Failure in High Dimensions\nThe argument above regarding the search time complexity of the algorithm fails in high dimensions. Let us elaborate why in this section.\nLet us accept that the assumptions that enabled the proof above hold and focus on G(d). The volume of a hypercube in d dimensions with sides that have length 2δâ is (2δâ)d. The volume of B(q, δâ) is Ïd/2δâd/Î (d/2+1), where Î denotes the Gamma function. For convenience, suppose that d is even, so that Î (d/2 + 1) = (d/2)!. As such G(d), the ratio between the two volumes, is:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 110 | G(d) = 2d(d/2)! Ïd/2 . (4.2)
Plugging this back into Equation (4.1), we arrive at:
(1+ Ga)" = (1 S(ay2y" ~ ou 2 OU)" = O(a)" d â)! 3)! dtl z ) )s
where in the third equality we used Stirlingâs formula, which approximates n! as
(5)! x Vmax L 4"? 1 asi = VF Gear * =O(d*).
The above shows that, the number of leaves that may be visited during the certification process has, asymptotically, an exponential dependency on d. That does not bode well for high dimensions.
There is an even simpler argument to make to show that in high dimensions the search algorithm must visit at least 2d data points during the certification
35
36
4 Branch-and-Bound Algorithms | 2401.09350#110 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 110,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "G(d) = 2d(d/2)! Ïd/2 . (4.2)\nPlugging this back into Equation (4.1), we arrive at:\n(1+ Ga)\" = (1 S(ay2y\" ~ ou 2 OU)\" = O(a)\" d â)! 3)! dtl z ) )s\nwhere in the third equality we used Stirlingâs formula, which approximates n! as\n(5)! x Vmax L 4\"? 1 asi = VF Gear * =O(d*).\nThe above shows that, the number of leaves that may be visited during the certification process has, asymptotically, an exponential dependency on d. That does not bode well for high dimensions.\nThere is an even simpler argument to make to show that in high dimensions the search algorithm must visit at least 2d data points during the certification\n35\n36\n4 Branch-and-Bound Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 111 | There is an even simpler argument to make to show that in high dimensions the search algorithm must visit at least 2d data points during the certification
35
36
4 Branch-and-Bound Algorithms
process. Our argument is as follows. We will show in Lemma 4.1 that, with high probability, the distance between the query point q and a randomly d. This implies that B(q, 뫉) has drawn data point concentrates sharply on a radius that is larger than 1 with high probability. Noting that the side of the unit hypercube is 2, it follows that B(q, 뫉) crosses decision boundaries across every dimension, making it necessary to visit the corresponding partitions for certification.
Finally, because each level of the tree splits on a single dimension, the reasoning above means that the certification process must visit â¦(d) levels of the tree. As a result, we visit at least 2â¦(d) data points. Of course, in high dimensions, we often have far fewer than 2d data points, so that we end up visiting every vector during certification. | 2401.09350#111 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 111,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "There is an even simpler argument to make to show that in high dimensions the search algorithm must visit at least 2d data points during the certification\n35\n36\n4 Branch-and-Bound Algorithms\nprocess. Our argument is as follows. We will show in Lemma 4.1 that, with high probability, the distance between the query point q and a randomly d. This implies that B(q, δâ) has drawn data point concentrates sharply on a radius that is larger than 1 with high probability. Noting that the side of the unit hypercube is 2, it follows that B(q, δâ) crosses decision boundaries across every dimension, making it necessary to visit the corresponding partitions for certification.\nFinally, because each level of the tree splits on a single dimension, the reasoning above means that the certification process must visit â¦(d) levels of the tree. As a result, we visit at least 2â¦(d) data points. Of course, in high dimensions, we often have far fewer than 2d data points, so that we end up visiting every vector during certification.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 112 | Lemma 4.1 The distance r between a randomly chosen point and its nearest neighbor among m points drawn uniformly at random from the unit hypercube is O(Vd/m'/4) with probability at least 1â O(1/2*). Proof. Consider the ball of radius r in d-dimensional unit hypercube with volume 1. Suppose, for notational convenience, that d is evenâwhether it is odd or even does not change our asymptotic conclusions. The volume of this ball is:
Ïd/2rd (d/2)! .
Since we have m points in the hypercube, the expected number of points that are contained in the ball of radius r is therefore:
Ïd/2rd (d/2)! m.
As a result, the radius r for which the ball contains one point in expectation is:
d/2.d ld neler d 1 (eo =)! (d/2â " GG) 1 od = = O(a).
Using Stirlingâs formula and letting Î consume the constants and small fac- tors completes the claim that r = | 2401.09350#112 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 112,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Lemma 4.1 The distance r between a randomly chosen point and its nearest neighbor among m points drawn uniformly at random from the unit hypercube is O(Vd/m'/4) with probability at least 1â O(1/2*). Proof. Consider the ball of radius r in d-dimensional unit hypercube with volume 1. Suppose, for notational convenience, that d is evenâwhether it is odd or even does not change our asymptotic conclusions. The volume of this ball is:\nÏd/2rd (d/2)! .\nSince we have m points in the hypercube, the expected number of points that are contained in the ball of radius r is therefore:\nÏd/2rd (d/2)! m.\nAs a result, the radius r for which the ball contains one point in expectation is:\nd/2.d ld neler d 1 (eo =)! (d/2â \" GG) 1 od = = O(a).\nUsing Stirlingâs formula and letting Î consume the constants and small fac- tors completes the claim that r =",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 113 | Using Stirlingâs formula and letting Î consume the constants and small fac- tors completes the claim that r =
d/m1/d. All that is left is bounding the probability of the event that r takes on the above value. For that, consider first the ball of radius r/2. The probability that this ball contains at least one point is at most 1/2d. To see this, note that the probability that a single point falls into this ball is:
n4/? (7 /2)4 1 qe/ 2d (d/2)! 24 (d/2)! ee 1/m
4.3 Randomized Trees
By the Union Bound, the probability that at least one point out of m points falls into this ball is at most m à 1/(m2d) = 1/2d. | 2401.09350#113 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 113,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Using Stirlingâs formula and letting Î consume the constants and small fac- tors completes the claim that r =\nd/m1/d. All that is left is bounding the probability of the event that r takes on the above value. For that, consider first the ball of radius r/2. The probability that this ball contains at least one point is at most 1/2d. To see this, note that the probability that a single point falls into this ball is:\nn4/? (7 /2)4 1 qe/ 2d (d/2)! 24 (d/2)! ee 1/m\n4.3 Randomized Trees\nBy the Union Bound, the probability that at least one point out of m points falls into this ball is at most m à 1/(m2d) = 1/2d.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 114 | By the Union Bound, the probability that at least one point out of m points falls into this ball is at most m à 1/(m2d) = 1/2d.
Next, consider the ball of radius 2r. The probability that it contains no points at all is at most (1 â 24/m)⢠~ exp(â2¢) < 1/2%, where we used the approximation that (1â 1/2)â © exp(â1) and the fact that exp(â2) < 1/z. To see why, it is enough to compute the probability that a single point does not fall into a ball of radius 2r, then by independence we arrive at the joint probability above. That probability is 1 minus the probability that the point falls into the ball, which is itself:
n4/2(27r)4 _ g Trt (d/2)! ~~ (d/2)!â Reger 1/m
hence the total probability (1 â 2d/m)m.
We have therefore shown that the probability that the distance of interest is r is extremely high and in the order of 1 â 1/2d, completing the proof. ââ
# 4.3 Randomized Trees | 2401.09350#114 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 114,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "By the Union Bound, the probability that at least one point out of m points falls into this ball is at most m à 1/(m2d) = 1/2d.\nNext, consider the ball of radius 2r. The probability that it contains no points at all is at most (1 â 24/m)⢠~ exp(â2¢) < 1/2%, where we used the approximation that (1â 1/2)â © exp(â1) and the fact that exp(â2) < 1/z. To see why, it is enough to compute the probability that a single point does not fall into a ball of radius 2r, then by independence we arrive at the joint probability above. That probability is 1 minus the probability that the point falls into the ball, which is itself:\nn4/2(27r)4 _ g Trt (d/2)! ~~ (d/2)!â Reger 1/m\nhence the total probability (1 â 2d/m)m.\nWe have therefore shown that the probability that the distance of interest is r is extremely high and in the order of 1 â 1/2d, completing the proof. ââ\n# 4.3 Randomized Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 115 | As we explained in Section 4.2, the search algorithm over a k-d Tree âindexâ consists of two operations: A single root-to-leaf traversal of the tree followed by backtracking to certify the candidate solution. As the analysis presented in the same section shows, it is the certification procedure that may need to visit virtually all data points. It is therefore not surprising that Liu et al. [2004] report that, in their experiments with low-dimensional vector collections (up to 30 dimensions), nearly 95% of the search time is spent in the latter phase. That observation naturally leads to the following question: What if we eliminated the certification step altogether? In other words, when given a query q, the search algorithm simply finds the cell that contains q in O(log m/mâ¦) time (where m = |X |), then returns the solution from among the m⦠vectors in that cellâa strategy Liu et al. [2004] call defeatist search. As Panigrahy [2008] shows for uniformly distributed vectors, however, the failure probability of the defeatist method is unacceptably high. That is pri- marily because, when a query | 2401.09350#115 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 115,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "As we explained in Section 4.2, the search algorithm over a k-d Tree âindexâ consists of two operations: A single root-to-leaf traversal of the tree followed by backtracking to certify the candidate solution. As the analysis presented in the same section shows, it is the certification procedure that may need to visit virtually all data points. It is therefore not surprising that Liu et al. [2004] report that, in their experiments with low-dimensional vector collections (up to 30 dimensions), nearly 95% of the search time is spent in the latter phase. That observation naturally leads to the following question: What if we eliminated the certification step altogether? In other words, when given a query q, the search algorithm simply finds the cell that contains q in O(log m/mâ¦) time (where m = |X |), then returns the solution from among the m⦠vectors in that cellâa strategy Liu et al. [2004] call defeatist search. As Panigrahy [2008] shows for uniformly distributed vectors, however, the failure probability of the defeatist method is unacceptably high. That is pri- marily because, when a query",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 116 | uniformly distributed vectors, however, the failure probability of the defeatist method is unacceptably high. That is pri- marily because, when a query is close to a decision boundary, the optimal solution may very well be on the other side. Figure 4.2(a) illustrates this phe- nomenon. As both the construction and search algorithms are deterministic, such a failure scenario is intrinsic to the algorithm and cannot be corrected once the tree has been constructed. Decision boundaries are hard and fast rules. | 2401.09350#116 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 116,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "uniformly distributed vectors, however, the failure probability of the defeatist method is unacceptably high. That is pri- marily because, when a query is close to a decision boundary, the optimal solution may very well be on the other side. Figure 4.2(a) illustrates this phe- nomenon. As both the construction and search algorithms are deterministic, such a failure scenario is intrinsic to the algorithm and cannot be corrected once the tree has been constructed. Decision boundaries are hard and fast rules.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 118 | (a) (b)
Fig. 4.2: Randomized construction of k-d Trees for a fixed collection of vectors (filled circles). Decision boundaries take random directions and are planted at a randomly-chosen point near the median. Repeating this procedure results in multiple âindexâ structures of the vector collection. Performing a âdefeatistâ search repeatedly for a given query (the empty circles) then leads to a higher probability of success.
regions as we split an internal node. As another example, we could place the decision boundaries at randomly chosen points close to the median, and have them take a randomly chosen direction. We illustrate the latter in Figure 4.2.
Such randomized decisions mean that, every time we construct a k-d Tree, we would obtain a different index of the data. Furthermore, by building a forest of randomized k-d Trees and repeating the defeatist search algorithm, we may be able to lower the failure probability!
These, as we will learn in this section, are indeed successful ideas that have been extensively explored in the literature [Liu et al., 2004, Ram and Sinha, 2019, Dasgupta and Sinha, 2015].
# 4.3.1 Randomized Partition Trees | 2401.09350#118 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 118,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "(a) (b) \nFig. 4.2: Randomized construction of k-d Trees for a fixed collection of vectors (filled circles). Decision boundaries take random directions and are planted at a randomly-chosen point near the median. Repeating this procedure results in multiple âindexâ structures of the vector collection. Performing a âdefeatistâ search repeatedly for a given query (the empty circles) then leads to a higher probability of success.\nregions as we split an internal node. As another example, we could place the decision boundaries at randomly chosen points close to the median, and have them take a randomly chosen direction. We illustrate the latter in Figure 4.2.\nSuch randomized decisions mean that, every time we construct a k-d Tree, we would obtain a different index of the data. Furthermore, by building a forest of randomized k-d Trees and repeating the defeatist search algorithm, we may be able to lower the failure probability!\nThese, as we will learn in this section, are indeed successful ideas that have been extensively explored in the literature [Liu et al., 2004, Ram and Sinha, 2019, Dasgupta and Sinha, 2015].\n# 4.3.1 Randomized Partition Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 119 | # 4.3.1 Randomized Partition Trees
Recall that a decision boundary in a k-d Tree is an axis-aligned hyperplane that is placed at the median point of the projection of data points onto a coordinate axis. Consider the following adjustment to that procedure, due originally to Liu et al. [2004] and further refined by Dasgupta and Sinha [2015]. Every time a node whose region is R is to be split, we first draw a random direction u by sampling a vector from the d-dimensional unit sphere, and a scalar β â [1/4, 3/4] uniformly at random. We then project all data points that are in R onto u, and obtain the β-fractile of the projections, θ. u together with θ form the decision boundary. We then proceed as before to partition the data points in R, by following the rule â¨u, v⩠⤠θ for every data point v â R. A node turns into a leaf if it contains a maximum of m⦠vectors.
4.3 Randomized Trees | 2401.09350#119 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 119,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 4.3.1 Randomized Partition Trees\nRecall that a decision boundary in a k-d Tree is an axis-aligned hyperplane that is placed at the median point of the projection of data points onto a coordinate axis. Consider the following adjustment to that procedure, due originally to Liu et al. [2004] and further refined by Dasgupta and Sinha [2015]. Every time a node whose region is R is to be split, we first draw a random direction u by sampling a vector from the d-dimensional unit sphere, and a scalar β â [1/4, 3/4] uniformly at random. We then project all data points that are in R onto u, and obtain the β-fractile of the projections, θ. u together with θ form the decision boundary. We then proceed as before to partition the data points in R, by following the rule â¨u, v⩠⤠θ for every data point v â R. A node turns into a leaf if it contains a maximum of m⦠vectors.\n4.3 Randomized Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 120 | 4.3 Randomized Trees
The procedure above gives us what Dasgupta and Sinha [2015] call a Ran- domized Partition (RP) Tree. You have already seen a visual demonstration of two RP Trees in Figure 4.2. Notice that, by requiring u to be a standard basis vector, fixing one u per each level of the tree, and letting β = 0.5, we reduce an RP Tree to the original k-d Tree.
What is the probability that a defeatist search over a single RP Tree fails to return the correct nearest neighbor? Dasgupta and Sinha [2015] proved that, that probability is related to the following potential function:
Igâ 2 Io 1 $(q,Â¥)=âS oi 4: (4, Â¥) my [quae Iaâ (4.3)
where m = |X |, Ï1 through Ïm are indices that sort data points by increasing distance to the query point q, so that x(Ï1) is the closest data point to q. | 2401.09350#120 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 120,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "4.3 Randomized Trees\nThe procedure above gives us what Dasgupta and Sinha [2015] call a Ran- domized Partition (RP) Tree. You have already seen a visual demonstration of two RP Trees in Figure 4.2. Notice that, by requiring u to be a standard basis vector, fixing one u per each level of the tree, and letting β = 0.5, we reduce an RP Tree to the original k-d Tree.\nWhat is the probability that a defeatist search over a single RP Tree fails to return the correct nearest neighbor? Dasgupta and Sinha [2015] proved that, that probability is related to the following potential function:\nIgâ 2 Io 1 $(q,Â¥)=âS oi 4: (4, Â¥) my [quae Iaâ (4.3)\nwhere m = |X |, Ï1 through Ïm are indices that sort data points by increasing distance to the query point q, so that x(Ï1) is the closest data point to q.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 121 | Notice that, a value of Φ that is close to 1 implies that nearly all data points are at the same distance from q. In that case, as we saw in Chap- ter 2, NN becomes unstable and approximate top-k retrieval becomes meaningless. When Φ is closer to 0, on the other hand, the optimal vector is well-separated from the rest of the collection.
Intuitively then, Φ is reflective of the difficulty or stability of the NN problem for a given query point. It makes sense then, that the probability of failure for q is related to this notion of difficulty of NN search: intuitively, when the nearest neighbor is far from other vectors, a defeatist search is more likely to yield the correct solution.
# 4.3.1.1 A Potential Function to Quantify the Difficulty of NN Search
Before we state the relationship between the failure probability and the po- tential function above more concretely, let us take a detour and understand where the expression for Φ comes from. All the arguments that we are about to make, including the lemmas and theorems, come directly from Dasgupta and Sinha [2015], though we repeat them here using our adopted notation for completeness. We also present an expanded proof of the formal results which follows the original proofs but elaborates on some of the steps. | 2401.09350#121 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 121,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Notice that, a value of Φ that is close to 1 implies that nearly all data points are at the same distance from q. In that case, as we saw in Chap- ter 2, NN becomes unstable and approximate top-k retrieval becomes meaningless. When Φ is closer to 0, on the other hand, the optimal vector is well-separated from the rest of the collection.\nIntuitively then, Φ is reflective of the difficulty or stability of the NN problem for a given query point. It makes sense then, that the probability of failure for q is related to this notion of difficulty of NN search: intuitively, when the nearest neighbor is far from other vectors, a defeatist search is more likely to yield the correct solution.\n# 4.3.1.1 A Potential Function to Quantify the Difficulty of NN Search\nBefore we state the relationship between the failure probability and the po- tential function above more concretely, let us take a detour and understand where the expression for Φ comes from. All the arguments that we are about to make, including the lemmas and theorems, come directly from Dasgupta and Sinha [2015], though we repeat them here using our adopted notation for completeness. We also present an expanded proof of the formal results which follows the original proofs but elaborates on some of the steps.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 122 | Let us start with a simplified setup where X consists of just two vectors x and y. Suppose that for a query point q, â¥q â xâ¥2 ⤠â¥q â yâ¥2. It turns out that, if we chose a random direction u and projected x and y onto it, then the probability that the projection of y onto u lands somewhere in between the projections of q and x onto u is a function of the potential function of Equation (4.3). The following lemma formalizes this relationship.
39
40
4 Branch-and-Bound Algorithms
Lemma 4.2 Suppose q, x, y â Rd and â¥q â xâ¥2 ⤠â¥q â yâ¥2. Let U â Sdâ1 be â a random direction and define y is between â q and
P[(G<9<%)V(E<9<O]= © arcsin 2 _ (d= %,y- 2) 2 = ares (20101 any) (Westy â as) ) | 2401.09350#122 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 122,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Let us start with a simplified setup where X consists of just two vectors x and y. Suppose that for a query point q, â¥q â xâ¥2 ⤠â¥q â yâ¥2. It turns out that, if we chose a random direction u and projected x and y onto it, then the probability that the projection of y onto u lands somewhere in between the projections of q and x onto u is a function of the potential function of Equation (4.3). The following lemma formalizes this relationship.\n39\n40\n4 Branch-and-Bound Algorithms\nLemma 4.2 Suppose q, x, y â Rd and â¥q â xâ¥2 ⤠â¥q â yâ¥2. Let U â Sdâ1 be â a random direction and define y is between â q and\nP[(G<9<%)V(E<9<O]= © arcsin 2 _ (d= %,y- 2) 2 = ares (20101 any) (Westy â as) )",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 123 | Proof. Assume, without loss of generality, that U is sampled from a d- dimensional standard Normal distribution: U â¼ N (0, Id). That assumption is inconsequential because normalizing U by its L2 norm gives a vector that lies on Sdâ1 as required. But because the norm of U does not affect the argument we need not explicitly perform the normalization.
Suppose further that we translate all vectors by q, and redefine q â 0, x â x â q, and y â y â q. We then rotate the vectors so that x = â¥xâ¥2e1, where e1 is the first standard basis vector. Neither the translation nor the rotation affects pairwise distances, and as such, no generality is lost due to these transformations.
Given this arrangement of vectors, it will be convenient to write U = (U1, U) and y = (y1, y) so as to make explicit the first coordinate of each vector (denoted by subscript 1) and the remaining coordinates (denoted by subscript ). | 2401.09350#123 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 123,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. Assume, without loss of generality, that U is sampled from a d- dimensional standard Normal distribution: U â¼ N (0, Id). That assumption is inconsequential because normalizing U by its L2 norm gives a vector that lies on Sdâ1 as required. But because the norm of U does not affect the argument we need not explicitly perform the normalization.\nSuppose further that we translate all vectors by q, and redefine q â 0, x â x â q, and y â y â q. We then rotate the vectors so that x = â¥xâ¥2e1, where e1 is the first standard basis vector. Neither the translation nor the rotation affects pairwise distances, and as such, no generality is lost due to these transformations.\nGiven this arrangement of vectors, it will be convenient to write U = (U1, U\u0001) and y = (y1, y\u0001) so as to make explicit the first coordinate of each vector (denoted by subscript 1) and the remaining coordinates (denoted by subscript \u0001).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 124 | It is safe to assume that y ̸= 0. The reason is that, if that were not the case, the two vectors x and y have an intrinsic dimensionality of 1 and are thus on a line. In that case, no matter which direction U we choose, â y will not fall between â x and â q = 0.
We can now write the probability of the event of interest as follows:
Pl (@<9<*)v(t<9<a)|= a E P[(0.< W,y) < |jnllal2) V ((lalla¥s < (W,y) <0)].
By expanding (U,y) = yiUi + (U,y1); it becomes clear that the ex- pression above measures the probability that (U, W1) falls in the interval (= ylUil, (llell2 â y1)|Ui|) when U, > 0 or (= (la |l2â 1)|Uil,ya|Ua)) oth- erwise. As such P[E] can be rewritten as follows: | 2401.09350#124 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 124,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "It is safe to assume that y\u0001 ̸= 0. The reason is that, if that were not the case, the two vectors x and y have an intrinsic dimensionality of 1 and are thus on a line. In that case, no matter which direction U we choose, â y will not fall between â x and â q = 0.\nWe can now write the probability of the event of interest as follows:\nPl (@<9<*)v(t<9<a)|= a E P[(0.< W,y) < |jnllal2) V ((lallaÂ¥s < (W,y) <0)].\nBy expanding (U,y) = yiUi + (U\u0001,y1); it becomes clear that the ex- pression above measures the probability that (U\u0001, W1) falls in the interval (= ylUil, (llell2 â y1)|Ui|) when U, > 0 or (= (la |l2â 1)|Uil,ya|Ua)) oth- erwise. As such P[E] can be rewritten as follows:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 125 | P(E] =P | - milUil < Wr.) < (lalla â yu) |Ua| | Ur = 0] P [Wr = 0) +P|- (lvll2 â wal < Cas wa) < wlM| | Ti < 0| P[U; <0}.
4.3 Randomized Trees
First, note that U1 is independent of U given that they are sampled from N (0, Id). Second, observe that â¨U, yâ© is distributed as N (0, â¥yâ¥2 2), which is symmetric, so that the two intervals have the same probability mass. These two observations simplify the expression above, so that P[E] becomes:
P(E] =P | âmlUil < W,2) < (ella â w 310A] =P[âulZ/< llnalle2â < (elle - 12] 2"| (- yi (Ia\I2 âH) | yall2â Wavalle 7]? P
where Z and Z â² are independent random variables drawn from N (0, 1).
Using the fact that the ratio of two independent Gaussian random variables follows a standard Cauchy distribution, we can calculate P[E] as follows: | 2401.09350#125 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 125,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "P(E] =P | - milUil < Wr.) < (lalla â yu) |Ua| | Ur = 0] P [Wr = 0) +P|- (lvll2 â wal < Cas wa) < wlM| | Ti < 0| P[U; <0}.\n4.3 Randomized Trees\nFirst, note that U1 is independent of U\u0001 given that they are sampled from N (0, Id). Second, observe that â¨U\u0001, y\u0001â© is distributed as N (0, â¥y\u0001â¥2 2), which is symmetric, so that the two intervals have the same probability mass. These two observations simplify the expression above, so that P[E] becomes:\nP(E] =P | âmlUil < W\u0001,2) < (ella â w 310A] =P[âulZ/< llnalle2â < (elle - 12] 2\"| (- yi (Ia\\I2 âH) \u0002| yall2â Wavalle 7]? P\nwhere Z and Z â² are independent random variables drawn from N (0, 1).\nUsing the fact that the ratio of two independent Gaussian random variables follows a standard Cauchy distribution, we can calculate P[E] as follows:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 126 | Using the fact that the ratio of two independent Gaussian random variables follows a standard Cauchy distribution, we can calculate P[E] as follows:
PIE] [orn dw vi/\ly\ille m(1+w*) 1 (Wel =u) ( yi ) = â |} arctan (| ââââ } â arctan | â T llylla ll~alle 1 x aretan {Halll T lylls â wllalla 1 ccin { lal Ills = vt arcsin z z |: 7 IIyll2 V lylls â 2ys Marla + Halle
In the third equality, we used the fact that arctan a + arctan b = arctan(a + b)/(1 â ab), and in the fourth equality we used the identity arctan a = 1 + a2. Substituting y1 = â¨y, xâ©/â¥xâ¥2 and noting that x and y arcsin a/ ââ have been shifted by q completes the proof.
Corollary 4.1 In the same configuration as in Lemma 4.2: | 2401.09350#126 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 126,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Using the fact that the ratio of two independent Gaussian random variables follows a standard Cauchy distribution, we can calculate P[E] as follows:\nPIE] [orn dw vi/\\ly\\ille m(1+w*) 1 (Wel =u) ( yi ) = â |} arctan (| ââââ } â arctan | â T lly\u0007lla ll~alle 1 x aretan {Halll T lylls â wllalla 1 ccin { lal Ills = vt arcsin z z |: 7 IIyll2 V lylls â 2ys Marla + Halle\nIn the third equality, we used the fact that arctan a + arctan b = arctan(a + b)/(1 â ab), and in the fourth equality we used the identity arctan a = 1 + a2. Substituting y1 = â¨y, xâ©/â¥xâ¥2 and noting that x and y arcsin a/ ââ have been shifted by q completes the proof.\nCorollary 4.1 In the same configuration as in Lemma 4.2:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 127 | Corollary 4.1 In the same configuration as in Lemma 4.2:
2o(a-temyh - (saa)? < lla â al2lly â alle Pi(q@<9<a)v(e<9<4)| < Hq, {x,y})
Proof. Applying the inequality θ ⥠sin θ ⥠2θ/Ï for 0 ⤠θ ⤠Ï/2 to ââ Lemma 4.2 implies the claim.
Now that we have examined the case of X = {x, y}, it is easy to extend the result to a configuration of m vectors.
41
42
4 Branch-and-Bound Algorithms
Theorem 4.1 Suppose q â Rd, X â Rd is a set of m vectors, and xâ â X the â nearest neighbor of q. Let U â Sdâ1 be a random direction, define v = â¨U, vâ©, and let â X = { â x | x â X }. Then:
Z Ey [fraction of X that is be Zz tween § and "| < =8(q,X). NlR | 2401.09350#127 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 127,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Corollary 4.1 In the same configuration as in Lemma 4.2:\n2o(a-temyh - (saa)? < lla â al2lly â alle Pi(q@<9<a)v(e<9<4)| < Hq, {x,y})\nProof. Applying the inequality θ ⥠sin θ ⥠2θ/Ï for 0 ⤠θ ⤠Ï/2 to ââ Lemma 4.2 implies the claim.\nNow that we have examined the case of X = {x, y}, it is easy to extend the result to a configuration of m vectors.\n41\n42\n4 Branch-and-Bound Algorithms\nTheorem 4.1 Suppose q â Rd, X â Rd is a set of m vectors, and xâ â X the â nearest neighbor of q. Let U â Sdâ1 be a random direction, define v = â¨U, vâ©, and let â X = { â x | x â X }. Then:\nZ Ey [fraction of X that is be Zz tween § and \"| < =8(q,X). NlR",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 128 | Z Ey [fraction of X that is be Zz tween § and "| < =8(q,X). NlR
Proof. Let Ï1 through Ïm be indices that order the elements of X by increas- ing distance to q, so that xâ = x(Ï1). Denote by Zi the event that â¨U, x(Ïi)â© falls between â xâ and â q. By Corollary 4.1:
P[Zi] ⤠1 2 â¥q â x(Ï1)â¥2 â¥q â x(Ïi)â¥2 .
We can now write the expectation of interest as follows:
m 1 i] < ~®(q, ¥). S59, %)
.
Corollary 4.2 Under the assumptions of Theorem 4.1, for any α â (0, 1) and any s-subset S of X that contains xâ:
Zz 4 1 P [at least a fraction of S is between q and x*] < Ia Ps(4 x), a
where:
=35 lqâ 2 lp a- 2] 2â
and Ï1 through Ïs are indices of the s vectors in X that are closest to q, ordered by increasing distance.
Proof. Apply Theorem 4.1 to the set S to obtain: | 2401.09350#128 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 128,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Z Ey [fraction of X that is be Zz tween § and \"| < =8(q,X). NlR\nProof. Let Ï1 through Ïm be indices that order the elements of X by increas- ing distance to q, so that xâ = x(Ï1). Denote by Zi the event that â¨U, x(Ïi)â© falls between â xâ and â q. By Corollary 4.1:\nP[Zi] ⤠1 2 â¥q â x(Ï1)â¥2 â¥q â x(Ïi)â¥2 .\nWe can now write the expectation of interest as follows:\nm 1 i] < ~®(q, Â¥). S59, %)\n.\nCorollary 4.2 Under the assumptions of Theorem 4.1, for any α â (0, 1) and any s-subset S of X that contains xâ:\nZz 4 1 P [at least a fraction of S is between q and x*] < Ia Ps(4 x), a\nwhere:\n=35 lqâ 2 lp a- 2] 2â\nand Ï1 through Ïs are indices of the s vectors in X that are closest to q, ordered by increasing distance.\nProof. Apply Theorem 4.1 to the set S to obtain:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 129 | Proof. Apply Theorem 4.1 to the set S to obtain:
Z Zz E[fraction of S that is between a and x*| < (9,5) < 5s(a.). Nile
Using Markovâs inequality (i.e., P[Z > α] ⤠E[Z]/α) completes the proof. ââ
The above is where the potential function of Equation (4.3) first emerges in its complete form for an arbitrary collection of vectors and its subsets. As we see, Φ bounds the expected number of vectors whose projection onto a random direction U falls between a query point and its nearest neighbor.
ââ
4.3 Randomized Trees
The reason this expected value is important (which subsequently justi- fies the importance of Φ) has to do with the fact that decision bound- aries are planted at some β-fractile point of the projections. As such, a bound on the number of points that fall between q and its nearest neighbor serves as a tool to bound the odds that the decision boundary may separate q from its nearest neighbor, which is the failure mode we wish to quantify.
# 4.3.1.2 Probability of Failure | 2401.09350#129 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 129,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. Apply Theorem 4.1 to the set S to obtain:\nZ Zz E[fraction of S that is between a and x*| < (9,5) < 5s(a.). Nile\nUsing Markovâs inequality (i.e., P[Z > α] ⤠E[Z]/α) completes the proof. ââ\nThe above is where the potential function of Equation (4.3) first emerges in its complete form for an arbitrary collection of vectors and its subsets. As we see, Φ bounds the expected number of vectors whose projection onto a random direction U falls between a query point and its nearest neighbor.\nââ\n4.3 Randomized Trees\nThe reason this expected value is important (which subsequently justi- fies the importance of Φ) has to do with the fact that decision bound- aries are planted at some β-fractile point of the projections. As such, a bound on the number of points that fall between q and its nearest neighbor serves as a tool to bound the odds that the decision boundary may separate q from its nearest neighbor, which is the failure mode we wish to quantify.\n# 4.3.1.2 Probability of Failure",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 130 | # 4.3.1.2 Probability of Failure
We are now ready to use Theorem 4.1 and Corollary 4.2 to derive the failure probability of the defeatist search over an RP Tree. To that end, notice that, the path from the root to a leaf is a sequence of log1/β(m/mâ¦) independent decisions that involve randomly projected data points. So if we were able to bound the failure probability of a single node, we can apply the union bound and obtain a bound on the failure probability of the tree. That is the intuition that leads to the following result.
Theorem 4.2 The probability that an RP Tree built for collection X of m vectors fails to find the nearest neighbor of a query q is at most:
L 2e S~F jin, 3 Gam
and where we use the shorthand ®,
with β = 3/4 and â = log1/β for Φs(q, X ).
Proof. Consider an internal node of the RP Tree that contains q and s data points including xâ, the nearest neighbor of q. If the decision boundary at this node separates q from xâ, then the defeatist search will fail. We therefore seek to quantify the probability of that event. | 2401.09350#130 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 130,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 4.3.1.2 Probability of Failure\nWe are now ready to use Theorem 4.1 and Corollary 4.2 to derive the failure probability of the defeatist search over an RP Tree. To that end, notice that, the path from the root to a leaf is a sequence of log1/β(m/mâ¦) independent decisions that involve randomly projected data points. So if we were able to bound the failure probability of a single node, we can apply the union bound and obtain a bound on the failure probability of the tree. That is the intuition that leads to the following result.\nTheorem 4.2 The probability that an RP Tree built for collection X of m vectors fails to find the nearest neighbor of a query q is at most:\nL 2e S~F jin, 3 Gam\nand where we use the shorthand ®,\nwith β = 3/4 and â = log1/β for Φs(q, X ).\nProof. Consider an internal node of the RP Tree that contains q and s data points including xâ, the nearest neighbor of q. If the decision boundary at this node separates q from xâ, then the defeatist search will fail. We therefore seek to quantify the probability of that event.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 131 | Denote by F the fraction of the s vectors that, once projected onto the random direction U associated with the node, fall between q and xâ. Recall that, the split threshold associated with the node is drawn uniformly from an interval of mass 1/2. As such, the probability that q is separated from xâ is at most F/(1/2). By integrating over F , we obtain:
43
44
4 Branch-and-Bound Algorithms
1 P lq is separated from x*| < [ P [F = f] is! 2 [Ptr > Aw < 2 [min (1, oat PD, /2 1 @ 2 | df + 2 | df 0 P,/2 2f 2e =@,
â. s "S,
The first equality uses the definition of expectation for a positive random variable, while the second inequality uses Corollary 4.2. Applying the union bound to a path from root to leaf, and noting that the size of the collection that falls into each node drops geometrically per level by a factor of at least ââ 3/4 completes the proof. | 2401.09350#131 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 131,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Denote by F the fraction of the s vectors that, once projected onto the random direction U associated with the node, fall between q and xâ. Recall that, the split threshold associated with the node is drawn uniformly from an interval of mass 1/2. As such, the probability that q is separated from xâ is at most F/(1/2). By integrating over F , we obtain:\n43\n44\n4 Branch-and-Bound Algorithms\n1 P lq is separated from x*| < [ P [F = f] is! 2 [Ptr > Aw < 2 [min (1, oat PD, /2 1 @ 2 | df + 2 | df 0 P,/2 2f 2e =@,\nâ. s \"S,\nThe first equality uses the definition of expectation for a positive random variable, while the second inequality uses Corollary 4.2. Applying the union bound to a path from root to leaf, and noting that the size of the collection that falls into each node drops geometrically per level by a factor of at least ââ 3/4 completes the proof.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 132 | We are thus able to express the failure probability as a function of Φ, a quantity that is defined for a particular q and a concrete collection of vectors. If we have a model of the data distribution, it may be possible to state more general bounds by bounding Φ itself. Dasgupta and Sinha [2015] demonstrate examples of this for two practical data distributions. Let us review one such example here.
# 4.3.1.3 Data Drawn from a Doubling Measure
Throughout our analysis of k-d Trees in Section 4.2, we considered the case where data points are uniformly distributed in Rd. As we argued in Chapter 3, in many practical situations, however, even though vectors are represented in Rd, they actually lie in some low-dimensional manifold with intrinsic dimen- sion d⦠where d⦠⪠d. This happens, for example, when data points are drawn from a doubling measure with low dimension as defined in Definition 3.1.
Dasgupta and Sinha [2015] prove that, if a collection of m vectors is sam- pled from a doubling measure with dimension dâ¦, then Φ can be bounded from above roughly by (1/m)1/d⦠. The following theorem presents their claim. | 2401.09350#132 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 132,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We are thus able to express the failure probability as a function of Φ, a quantity that is defined for a particular q and a concrete collection of vectors. If we have a model of the data distribution, it may be possible to state more general bounds by bounding Φ itself. Dasgupta and Sinha [2015] demonstrate examples of this for two practical data distributions. Let us review one such example here.\n# 4.3.1.3 Data Drawn from a Doubling Measure\nThroughout our analysis of k-d Trees in Section 4.2, we considered the case where data points are uniformly distributed in Rd. As we argued in Chapter 3, in many practical situations, however, even though vectors are represented in Rd, they actually lie in some low-dimensional manifold with intrinsic dimen- sion d⦠where d⦠⪠d. This happens, for example, when data points are drawn from a doubling measure with low dimension as defined in Definition 3.1.\nDasgupta and Sinha [2015] prove that, if a collection of m vectors is sam- pled from a doubling measure with dimension dâ¦, then Φ can be bounded from above roughly by (1/m)1/d⦠. The following theorem presents their claim.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 133 | Theorem 4.3 Suppose a collection X of m vectors is drawn from µ, a con- tinuous, doubling measure on Rd with dimension d⦠⥠2. For an arbitrary δ â (0, 1/2), with probability at least 1 â 3δ, for all 2 ⤠s ⤠m:
4.3 Randomized Trees
2 1"" os) <6( 25) .
Using the result above, Dasgupta and Sinha [2015] go on to prove that, under the same conditions, with probability at least 1 â 3δ, the failure prob- ability of an RP Tree is bounded above by:
Mo ; 5 1/do cold 4 ham) (Smeal )
where c⦠is an absolute constant, and m⦠⥠câ¦3d⦠max(1, ln 1/δ).
The results above tell us that, so long as the space has a small intrinsic dimension, we can make the probability of failing to find the optimal solution arbitrarily small.
# 4.3.2 Spill Trees | 2401.09350#133 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 133,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Theorem 4.3 Suppose a collection X of m vectors is drawn from µ, a con- tinuous, doubling measure on Rd with dimension d⦠⥠2. For an arbitrary δ â (0, 1/2), with probability at least 1 â 3δ, for all 2 ⤠s ⤠m:\n4.3 Randomized Trees\n2 1\"\" os) <6( 25) .\nUsing the result above, Dasgupta and Sinha [2015] go on to prove that, under the same conditions, with probability at least 1 â 3δ, the failure prob- ability of an RP Tree is bounded above by:\nMo ; 5 1/do cold 4 ham) (Smeal )\nwhere c⦠is an absolute constant, and m⦠⥠câ¦3d⦠max(1, ln 1/δ).\nThe results above tell us that, so long as the space has a small intrinsic dimension, we can make the probability of failing to find the optimal solution arbitrarily small.\n# 4.3.2 Spill Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 134 | # 4.3.2 Spill Trees
The Spill Tree [Liu et al., 2004] is another randomized variant of the k-d Tree. The algorithm to construct a Spill Tree comes with a hyperparameter α â [0, 1/2] that is typically a small constant closer to 0. Given an α, the Spill Tree modifies the tree construction algorithm of the k-d Tree as follows. When splitting a node whose region is R, we first project all vectors contained in R onto a random direction U , then find the median of the resulting distribution. However, instead of partitioning the vectors based on which side of the median they are on, the algorithm forms two overlapping sets. The âleftâ set contains all vectors in R whose projection onto U is smaller than the (1/2 + α)-fractile point of the distribution, while the ârightâ set consists of those that fall to the right of the (1/2 â α)-fractile point. As before, a node becomes a leaf when it has a maximum of m⦠vectors. | 2401.09350#134 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 134,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 4.3.2 Spill Trees\nThe Spill Tree [Liu et al., 2004] is another randomized variant of the k-d Tree. The algorithm to construct a Spill Tree comes with a hyperparameter α â [0, 1/2] that is typically a small constant closer to 0. Given an α, the Spill Tree modifies the tree construction algorithm of the k-d Tree as follows. When splitting a node whose region is R, we first project all vectors contained in R onto a random direction U , then find the median of the resulting distribution. However, instead of partitioning the vectors based on which side of the median they are on, the algorithm forms two overlapping sets. The âleftâ set contains all vectors in R whose projection onto U is smaller than the (1/2 + α)-fractile point of the distribution, while the ârightâ set consists of those that fall to the right of the (1/2 â α)-fractile point. As before, a node becomes a leaf when it has a maximum of m⦠vectors.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 135 | During search, the algorithm performs a defeatist search by routing the query point q based on a comparison of its projection onto the random direc- tion associated with each node, and the median point. It is clear that with this strategy, if the nearest neighbor of q is close to the decision boundary of a node, we do not increase the likelihood of failure whether we route q to the left child or to the right one. Figure 4.3 shows an example of the defeatist search over a Spill Tree.
45
46
4 Branch-and-Bound Algorithms | 2401.09350#135 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 135,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "During search, the algorithm performs a defeatist search by routing the query point q based on a comparison of its projection onto the random direc- tion associated with each node, and the median point. It is clear that with this strategy, if the nearest neighbor of q is close to the decision boundary of a node, we do not increase the likelihood of failure whether we route q to the left child or to the right one. Figure 4.3 shows an example of the defeatist search over a Spill Tree.\n45\n46\n4 Branch-and-Bound Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 136 | Fig. 4.3: Defeatist search over a Spill Tree. In a Spill Tree, vectors that are close to the decision boundary are, in effect, duplicated, with their copy âspillingâ over to the other side of the boundary. This is depicted for a few example regions as the blue shaded area that straddles the decision boundary: vectors that fall into the shaded area belong to neighboring regions. For example, regions G and H share two vectors. As such, a defeatist search for the example query (the empty circle) looks through not just the region E but its extended region that overlaps with F .
# 4.3.2.1 Space Overhead
One obvious downside of the Spill Tree is that a single data point may end up in multiple leaf nodes, which increases the space complexity. We can quantify that by noting that the depth of the tree on a collection of m vectors is at most log1/(1/2+α)(m/mâ¦), so that the total number of vectors in all leaves is:
. â¢. \ log a2 M y1/(1-log(1+2a Mo2Br/a/2ztay(m/mo) â mo( ) 1/(A/2ta) 2 o( ) /( ( ) Mo Mo
. | 2401.09350#136 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 136,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 4.3: Defeatist search over a Spill Tree. In a Spill Tree, vectors that are close to the decision boundary are, in effect, duplicated, with their copy âspillingâ over to the other side of the boundary. This is depicted for a few example regions as the blue shaded area that straddles the decision boundary: vectors that fall into the shaded area belong to neighboring regions. For example, regions G and H share two vectors. As such, a defeatist search for the example query (the empty circle) looks through not just the region E but its extended region that overlaps with F .\n# 4.3.2.1 Space Overhead\nOne obvious downside of the Spill Tree is that a single data point may end up in multiple leaf nodes, which increases the space complexity. We can quantify that by noting that the depth of the tree on a collection of m vectors is at most log1/(1/2+α)(m/mâ¦), so that the total number of vectors in all leaves is:\n. â¢. \\ log a2 M y1/(1-log(1+2a Mo2Br/a/2ztay(m/mo) â mo( ) 1/(A/2ta) 2 o( ) /( ( ) Mo Mo\n.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 137 | .
As such, the space complexity of a Spill Tree is O(m1/(1âlog(1+2α))).
# 4.3.2.2 Probability of Failure
The defeatist search over a Spill Tree fails to return the nearest neighbor xâ if the following event takes place at any of the nodes that contains q and xâ. That is the event where the projections of q and xâ are separated by the median and where the projection of xâ is separated from the median by at least α-fraction of the vectors. That event happens when the projections of q and xâ are separated by at least α-fraction of the vectors in some node along the path.
The probability of the event above can be bounded by Corollary 4.2. By applying the union bound to a root-to-leaf path, and noting that the size of the collection reduces at each level by a factor of at least 1/2 + α, we obtain the following result:
4.4 Cover Trees
Theorem 4.4 The probability that a Spill Tree built for collection X of m vectors fails to find the nearest neighbor of a query q is at most:
â4 Ye Pm (a Â¥), (OL l=0
(m/m.).
with β = 1/2 + α and â = log1/β
# 4.4 Cover Trees | 2401.09350#137 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 137,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": ".\nAs such, the space complexity of a Spill Tree is O(m1/(1âlog(1+2α))).\n# 4.3.2.2 Probability of Failure\nThe defeatist search over a Spill Tree fails to return the nearest neighbor xâ if the following event takes place at any of the nodes that contains q and xâ. That is the event where the projections of q and xâ are separated by the median and where the projection of xâ is separated from the median by at least α-fraction of the vectors. That event happens when the projections of q and xâ are separated by at least α-fraction of the vectors in some node along the path.\nThe probability of the event above can be bounded by Corollary 4.2. By applying the union bound to a root-to-leaf path, and noting that the size of the collection reduces at each level by a factor of at least 1/2 + α, we obtain the following result:\n4.4 Cover Trees\nTheorem 4.4 The probability that a Spill Tree built for collection X of m vectors fails to find the nearest neighbor of a query q is at most:\nâ4 Ye Pm (a Â¥), (OL l=0\n(m/m.).\nwith β = 1/2 + α and â = log1/β\n# 4.4 Cover Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 138 | (m/m.).
with β = 1/2 + α and â = log1/β
# 4.4 Cover Trees
The branch-and-bound algorithms we have reviewed thus far divide a col- lection recursively into exactly two sub-collections, using a hyperplane as a decision boundary. Some also have a certification process that involves back- tracking from a leaf node whose region contains a query to the root node. As we noted in Section 4.1, however, none of these choices is absolutely nec- essary. In fact, branching and bounding can be done entirely differently. We review in this section a popular example that deviates from that pattern, a data structure known as the Cover Tree [Beygelzimer et al., 2006].
It is more intuitive to describe the Cover Tree, as well as the construction and search algorithms over it, in the abstract first. This is what Beygelzimer et al. [2006] call the implicit representation. Let us first describe its structure, then review its properties and explain the relevant algorithms, and only then discuss how the abstract tree can be implemented concretely.
# 4.4.1 The Abstract Cover Tree and its Properties | 2401.09350#138 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 138,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "(m/m.).\nwith β = 1/2 + α and â = log1/β\n# 4.4 Cover Trees\nThe branch-and-bound algorithms we have reviewed thus far divide a col- lection recursively into exactly two sub-collections, using a hyperplane as a decision boundary. Some also have a certification process that involves back- tracking from a leaf node whose region contains a query to the root node. As we noted in Section 4.1, however, none of these choices is absolutely nec- essary. In fact, branching and bounding can be done entirely differently. We review in this section a popular example that deviates from that pattern, a data structure known as the Cover Tree [Beygelzimer et al., 2006].\nIt is more intuitive to describe the Cover Tree, as well as the construction and search algorithms over it, in the abstract first. This is what Beygelzimer et al. [2006] call the implicit representation. Let us first describe its structure, then review its properties and explain the relevant algorithms, and only then discuss how the abstract tree can be implemented concretely.\n# 4.4.1 The Abstract Cover Tree and its Properties",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 139 | # 4.4.1 The Abstract Cover Tree and its Properties
The abstract Cover Tree is a tree structure with infinite depth that is defined for a proper metric δ(·, ·). Each level of the tree is numbered by an integer that starts from â at the level of the root node and decrements, to ââ, at each subsequent level. Each node represents a single data point. If we denote the collection of nodes on level â by Câ, then Câ is a set, in the sense that the data points represented by those nodes are distinct. But Câ â Cââ1, so that once a node appears in level â, it necessarily appears in levels (â â 1) onward. That implies that, in the abstract Cover Tree, Câ contains a single data point, and Cââ = X is the entire collection.
This structure, which is illustrated in Figure 4.4 for an example collection of vectors, obeys three invariants. That is, all algorithms that construct the tree or manipulate it in any way must guarantee that the three properties are not violated. These invariants are: ⢠Nesting: As we noted, Câ â Cââ1.
47
48
4 Branch-and-Bound Algorithms
b= eat £20 | 2401.09350#139 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 139,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 4.4.1 The Abstract Cover Tree and its Properties\nThe abstract Cover Tree is a tree structure with infinite depth that is defined for a proper metric δ(·, ·). Each level of the tree is numbered by an integer that starts from â at the level of the root node and decrements, to ââ, at each subsequent level. Each node represents a single data point. If we denote the collection of nodes on level â by Câ, then Câ is a set, in the sense that the data points represented by those nodes are distinct. But Câ â Cââ1, so that once a node appears in level â, it necessarily appears in levels (â â 1) onward. That implies that, in the abstract Cover Tree, Câ contains a single data point, and Cââ = X is the entire collection.\nThis structure, which is illustrated in Figure 4.4 for an example collection of vectors, obeys three invariants. That is, all algorithms that construct the tree or manipulate it in any way must guarantee that the three properties are not violated. These invariants are: ⢠Nesting: As we noted, Câ â Cââ1.\n47\n48\n4 Branch-and-Bound Algorithms\nb= eat £20",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 140 | 47
48
4 Branch-and-Bound Algorithms
b= eat £20
Fig. 4.4: Illustration of the abstract Cover Tree for a collection of 8 vectors. Nodes on level â of the tree are separated by at least 2â by the separation invariant. Nodes on level â cover nodes on level (â â 1) with a ball of radius at most 2â by the covering invariant. Once a node appears in the tree, it will appear on all subsequent levels as its own child (solid arrows), by the nesting invariant.
# Algorithm 1: Nearest Neighbor search over a Cover Tree.
Input: Cover Tree with metric δ(·, ·); query point q. Result: Exact NN of q. 1: Qâ â Câ ; â· Câ is the set of nodes on level â 2: for â from â to ââ do 3:
Q â {Children(v) | v â Qâ} ; â· Children(·) returns the children of its argument. Qââ1 â {u | δ(q, u) ⤠δ(q, Q) + 2â} ; ⷠδ(u, S) â minvâS δ(u, v). | 2401.09350#140 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 140,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "47\n48\n4 Branch-and-Bound Algorithms\nb= eat £20\nFig. 4.4: Illustration of the abstract Cover Tree for a collection of 8 vectors. Nodes on level â of the tree are separated by at least 2â by the separation invariant. Nodes on level â cover nodes on level (â â 1) with a ball of radius at most 2â by the covering invariant. Once a node appears in the tree, it will appear on all subsequent levels as its own child (solid arrows), by the nesting invariant.\n# Algorithm 1: Nearest Neighbor search over a Cover Tree.\nInput: Cover Tree with metric δ(·, ·); query point q. Result: Exact NN of q. 1: Qâ â Câ ; â· Câ is the set of nodes on level â 2: for â from â to ââ do 3:\nQ â {Children(v) | v â Qâ} ; â· Children(·) returns the children of its argument. Qââ1 â {u | δ(q, u) ⤠δ(q, Q) + 2â} ; ⷠδ(u, S) â minvâS δ(u, v).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 141 | 4: 5: end for 6: return arg minuâQââ
# δ(q, u)
⢠Covering: For every node u â Cââ1 there is a node v â Câ such that δ(u, v) < 2â. In other words, every node in the next level (â â 1) of the tree is âcoveredâ by an open ball of radius 2â around a node in the current level, â.
Separation: All nodes on the same level â are separated by a distance of
at least 2â. Formally, if u, v â Câ, then δ(u, v) > 2â.
# 4.4.2 The Search Algorithm
We have seen what a Cover Tree looks like and what properties it is guaran- teed to maintain. Given this structure, how do we find the nearest neighbor of a query point? That turns out to be a fairly simple algorithm as shown in Algorithm 1.
4.4 Cover Trees | 2401.09350#141 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 141,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "4: 5: end for 6: return arg minuâQââ\n# δ(q, u)\n⢠Covering: For every node u â Cââ1 there is a node v â Câ such that δ(u, v) < 2â. In other words, every node in the next level (â â 1) of the tree is âcoveredâ by an open ball of radius 2â around a node in the current level, â.\nSeparation: All nodes on the same level â are separated by a distance of\nat least 2â. Formally, if u, v â Câ, then δ(u, v) > 2â.\n# 4.4.2 The Search Algorithm\nWe have seen what a Cover Tree looks like and what properties it is guaran- teed to maintain. Given this structure, how do we find the nearest neighbor of a query point? That turns out to be a fairly simple algorithm as shown in Algorithm 1.\n4.4 Cover Trees",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 142 | 4.4 Cover Trees
Algorithm 1 always maintains a current set of candidates in Qâ as it visits level â of the tree. In each iteration of the loop on Line 2, it creates a tem- porary setâdenoted by Qâby collecting the children of all nodes in Qâ. It then prunes the nodes in Q based on the condition on Line 4. Eventually, the algorithm returns the exact nearest neighbor of query q by performing exhaustive search over the nodes in Qââ.
Let us understand why the algorithm is correct. In a way, it is enough to argue that the pruning condition on Line 4 never discards an ancestor of the nearest neighbor. If that were the case, we are done proving the correctness of the algorithm: Qââ is guaranteed to have the nearest neighbor, at which point we will find it on Line 6. | 2401.09350#142 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 142,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "4.4 Cover Trees\nAlgorithm 1 always maintains a current set of candidates in Qâ as it visits level â of the tree. In each iteration of the loop on Line 2, it creates a tem- porary setâdenoted by Qâby collecting the children of all nodes in Qâ. It then prunes the nodes in Q based on the condition on Line 4. Eventually, the algorithm returns the exact nearest neighbor of query q by performing exhaustive search over the nodes in Qââ.\nLet us understand why the algorithm is correct. In a way, it is enough to argue that the pruning condition on Line 4 never discards an ancestor of the nearest neighbor. If that were the case, we are done proving the correctness of the algorithm: Qââ is guaranteed to have the nearest neighbor, at which point we will find it on Line 6.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 143 | The fact that Algorithm 1 never prunes the ancestor of the solution is easy to establish. To see how, consider the distance between u ⬠Ce_; and any of its descendants, v. The distance between the two vectors is bounded as follows: d(u,v) < 4 2! â 2°. Furthermore, because 6 is proper, by triangle inequality, we know that: 6(q,u*) < 6(¢,Q) + 6(Q,u*), where u* is the solution and a descendant of wu ⬠Cy_1. As such, any candidate whose distance is greater than 5(q, Q)+6(Q, u*) < 6(q,Q)+2° can be safely pruned.
The search algorithm has an ϵ-approximate variant too. To obtain a solu- tion that is at most (1 + ϵ)δ(q, uâ) away from q, assuming uâ is the optimal solution, we need only to change the termination condition on Line 2, by exiting the loop as soon as δ(q, Qâ) ⥠2â+1(1 + 1/ϵ). Let us explain why the resulting algorithm is correct. | 2401.09350#143 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 143,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The fact that Algorithm 1 never prunes the ancestor of the solution is easy to establish. To see how, consider the distance between u ⬠Ce_; and any of its descendants, v. The distance between the two vectors is bounded as follows: d(u,v) < 4 2! â 2°. Furthermore, because 6 is proper, by triangle inequality, we know that: 6(q,u*) < 6(¢,Q) + 6(Q,u*), where u* is the solution and a descendant of wu ⬠Cy_1. As such, any candidate whose distance is greater than 5(q, Q)+6(Q, u*) < 6(q,Q)+2° can be safely pruned.\nThe search algorithm has an ϵ-approximate variant too. To obtain a solu- tion that is at most (1 + ϵ)δ(q, uâ) away from q, assuming uâ is the optimal solution, we need only to change the termination condition on Line 2, by exiting the loop as soon as δ(q, Qâ) ⥠2â+1(1 + 1/ϵ). Let us explain why the resulting algorithm is correct.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 144 | Suppose that the algorithm terminates early when it reaches level â. That means that 2â+1(1 + 1/ϵ) ⤠δ(q, Qâ). We have already seen that δ(q, Qâ) ⤠2â+1, and by triangle inequality, that δ(q, Qâ) ⤠δ(q, uâ) + 2â+1. So we have bounded δ(q, Qâ) from below and above, resulting in the following inequality:
1 gett (1 + -) < 5(qut) +2! => 2! < 66(q,u*). â¬
Putting all that together, we have shown that δ(q, Qâ) ⤠(1 + ϵ)δ(q, uâ), so that Line 6 returns an ϵ-approximate solution.
# 4.4.3 The Construction Algorithm
Inserting a single vector into the Cover Tree âindexâ is a procedure that is similar to the search algorithm but is better conceptualized recursively, as shown in Algorithm 2. | 2401.09350#144 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 144,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Suppose that the algorithm terminates early when it reaches level â. That means that 2â+1(1 + 1/ϵ) ⤠δ(q, Qâ). We have already seen that δ(q, Qâ) ⤠2â+1, and by triangle inequality, that δ(q, Qâ) ⤠δ(q, uâ) + 2â+1. So we have bounded δ(q, Qâ) from below and above, resulting in the following inequality:\n1 gett (1 + -) < 5(qut) +2! => 2! < 66(q,u*). â¬\nPutting all that together, we have shown that δ(q, Qâ) ⤠(1 + ϵ)δ(q, uâ), so that Line 6 returns an ϵ-approximate solution.\n# 4.4.3 The Construction Algorithm\nInserting a single vector into the Cover Tree âindexâ is a procedure that is similar to the search algorithm but is better conceptualized recursively, as shown in Algorithm 2.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 145 | Inserting a single vector into the Cover Tree âindexâ is a procedure that is similar to the search algorithm but is better conceptualized recursively, as shown in Algorithm 2.
It is important to note that the procedure in Algorithm 2 assumes that the point p is not present in the tree. That is a harmless assumption as the existence of p can be checked by a simple invocation of Algorithm 1. We can therefore safely assume that δ(p, Q) for any Q formed on Line 1 is
49
50
4 Branch-and-Bound Algorithms
# Algorithm 2: Insertion of a vector into a Cover Tree.
Input: Cover Tree T with metric δ(·, ·); New Vector p; Level â; Candidate set Qâ. Result: Cover Tree containing p. 1: Q â {Children(u) | u â Qâ} 2: if δ(p, Q) > 2â then 3: return â·â 4: else 5: 6: 7: 8: 9: 10: 11: 12: 13: end if | 2401.09350#145 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 145,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Inserting a single vector into the Cover Tree âindexâ is a procedure that is similar to the search algorithm but is better conceptualized recursively, as shown in Algorithm 2.\nIt is important to note that the procedure in Algorithm 2 assumes that the point p is not present in the tree. That is a harmless assumption as the existence of p can be checked by a simple invocation of Algorithm 1. We can therefore safely assume that δ(p, Q) for any Q formed on Line 1 is\n49\n50\n4 Branch-and-Bound Algorithms\n# Algorithm 2: Insertion of a vector into a Cover Tree.\nInput: Cover Tree T with metric δ(·, ·); New Vector p; Level â; Candidate set Qâ. Result: Cover Tree containing p. 1: Q â {Children(u) | u â Qâ} 2: if δ(p, Q) > 2â then 3: return â·â 4: else 5: 6: 7: 8: 9: 10: 11: 12: 13: end if",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 146 | strictly positive. That assumption guarantees that the algorithm eventually terminates. That is because δ(p, Q) > 0 so that ultimately we will invoke the algorithm with a value â such that δ(p, Q) > 2â, at which point Line 2 terminates the recursion.
We can also see why Line 6 is bound to evaluate to True at some point during the execution of the algorithm. That is because there must exist a level â such that 2ââ1 < δ(p, Q) ⤠2â. That implies that the point p will ultimately be inserted into the tree.
What about the three invariants of the Cover Tree? We must now show that the resulting tree maintains those properties: nesting, covering, and sepa- ration. The covering invariant is immediately guaranteed as a result of Line 6. The nesting invariant too is trivially maintained because we can insert p as its own child for all subsequent levels. | 2401.09350#146 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 146,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "strictly positive. That assumption guarantees that the algorithm eventually terminates. That is because δ(p, Q) > 0 so that ultimately we will invoke the algorithm with a value â such that δ(p, Q) > 2â, at which point Line 2 terminates the recursion.\nWe can also see why Line 6 is bound to evaluate to True at some point during the execution of the algorithm. That is because there must exist a level â such that 2ââ1 < δ(p, Q) ⤠2â. That implies that the point p will ultimately be inserted into the tree.\nWhat about the three invariants of the Cover Tree? We must now show that the resulting tree maintains those properties: nesting, covering, and sepa- ration. The covering invariant is immediately guaranteed as a result of Line 6. The nesting invariant too is trivially maintained because we can insert p as its own child for all subsequent levels.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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