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2401.09350 | 147 | What remains is to show that the insertion algorithm maintains the sep- aration property. To that end, suppose p has been inserted into Cââ1 and consider its sibling u â Cââ1. If u â Q, then it is clear that δ(p, u) > 2ââ1 because Line 6 must have evaluated to True. On the other hand, if u /â Q, that means that there was some ââ² > â where some ancestor of u, uâ² â Cââ²â1, was pruned on Line 5, so that δ(p, uâ²) > 2ââ² . Using the covering invariant, we can deduce that:
£ 5(p,u) > 5(p,u') â Ss 2! l=¢'-1 = (pw) â (2 = 2°) > 2â â (2 â 2%) = 28.
4.5 Closing Remarks
That concludes the proof that δ(p, Cââ1) > 2ââ1, showing that Algorithm 2 maintains the separation invariant.
# 4.4.4 The Concrete Cover Tree | 2401.09350#147 | 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": 147,
"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 remains is to show that the insertion algorithm maintains the sep- aration property. To that end, suppose p has been inserted into Cââ1 and consider its sibling u â Cââ1. If u â Q, then it is clear that δ(p, u) > 2ââ1 because Line 6 must have evaluated to True. On the other hand, if u /â Q, that means that there was some ââ² > â where some ancestor of u, uâ² â Cââ²â1, was pruned on Line 5, so that δ(p, uâ²) > 2ââ² . Using the covering invariant, we can deduce that:\n£ 5(p,u) > 5(p,u') â Ss 2! l=¢'-1 = (pw) â (2 = 2°) > 2â â (2 â 2%) = 28.\n4.5 Closing Remarks\nThat concludes the proof that δ(p, Cââ1) > 2ââ1, showing that Algorithm 2 maintains the separation invariant.\n# 4.4.4 The Concrete Cover Tree",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 148 | # 4.4.4 The Concrete Cover Tree
The abstract tree we described earlier has infinite depth. While that repre- sentation is convenient for explaining the data structure and algorithms that operate on it, it is not practical. But it is easy to derive a concrete instance of the data structure, without changing the algorithmic details, to obtain what Beygelzimer et al. [2006] call the explicit representation.
One straightforward way of turning the abstract Cover Tree into a concrete one is by turning a node into a (terminal) leaf if it is its only childârecall that, a node in the abstract Cover Tree is its own child, indefinitely. For example, in Figure 4.4, all nodes on level 0 would become leaves and the Cover Tree would end at that depth. We leave it as an exercise to show that the concrete representation of the tree does not affect the correctness of Algorithms 1 and 2. | 2401.09350#148 | 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": 148,
"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": "# 4.4.4 The Concrete Cover Tree\nThe abstract tree we described earlier has infinite depth. While that repre- sentation is convenient for explaining the data structure and algorithms that operate on it, it is not practical. But it is easy to derive a concrete instance of the data structure, without changing the algorithmic details, to obtain what Beygelzimer et al. [2006] call the explicit representation.\nOne straightforward way of turning the abstract Cover Tree into a concrete one is by turning a node into a (terminal) leaf if it is its only childârecall that, a node in the abstract Cover Tree is its own child, indefinitely. For example, in Figure 4.4, all nodes on level 0 would become leaves and the Cover Tree would end at that depth. We leave it as an exercise to show that the concrete representation of the tree does not affect the correctness of Algorithms 1 and 2.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 149 | The concrete form is not only important for making the data structure practical, it is also necessary for analysis. For example, as Beygelzimer et al. [2006] prove that the space complexity of the concrete Cover Tree is O(m) with m = |X |, whereas the abstract form is infinitely large. The time com- plexity of the insertion and search algorithms also use the concrete form, but they further require assumptions on the data distribution. Beygelzimer et al. [2006] present their analysis for vectors that are drawn from a doubling mea- sure, as we have defined in Definition 3.1. However, their claims have been disputed [Curtin, 2016] by counter-examples [Elkin and Kurlin, 2022], and corrected in a recent work [Elkin and Kurlin, 2023].
# 4.5 Closing Remarks
This chapter has only covered algorithms that convey the foundations of a branch-and-bound approach to NN search. Indeed, we left out a number of alternative constructions that are worth mentioning as we close this chapter.
# 4.5.1 Alternative Constructions and Extensions
The standard k-d Tree itself, as an example, can be instantiated by using a different splitting procedure, such as splitting on the axis along which the data exhibits the greatest spread. PCA Trees [Sproull, 1991], PAC Trees [Mc51
52 | 2401.09350#149 | 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": 149,
"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 concrete form is not only important for making the data structure practical, it is also necessary for analysis. For example, as Beygelzimer et al. [2006] prove that the space complexity of the concrete Cover Tree is O(m) with m = |X |, whereas the abstract form is infinitely large. The time com- plexity of the insertion and search algorithms also use the concrete form, but they further require assumptions on the data distribution. Beygelzimer et al. [2006] present their analysis for vectors that are drawn from a doubling mea- sure, as we have defined in Definition 3.1. However, their claims have been disputed [Curtin, 2016] by counter-examples [Elkin and Kurlin, 2022], and corrected in a recent work [Elkin and Kurlin, 2023].\n# 4.5 Closing Remarks\nThis chapter has only covered algorithms that convey the foundations of a branch-and-bound approach to NN search. Indeed, we left out a number of alternative constructions that are worth mentioning as we close this chapter.\n# 4.5.1 Alternative Constructions and Extensions\nThe standard k-d Tree itself, as an example, can be instantiated by using a different splitting procedure, such as splitting on the axis along which the data exhibits the greatest spread. PCA Trees [Sproull, 1991], PAC Trees [Mc51\n52",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 150 | 52
4 Branch-and-Bound Algorithms
Names, 2001], and Max-Margin Trees [Ram et al., 2012] offer other ways of choosing the axis or direction along which the algorithm partitions the data. Vantage-point Trees [Yianilos, 1993], as another example, follow the same iterative procedure as k-d Trees, but partition the space using hyperspheres rather than hyperplanes.
There are also various other randomized constructions of tree index struc- tures for NN search. Panigrahy [2008], for instance, construct a standard k-d Tree over the original data points but, during search, perturb the query point. Repeating the perturb-then-search scheme reduces the failure probability of a defeatist search over the k-d Tree.
Sinha [2014] proposes a different variant of the RP Tree where, instead of a random projection, they choose the principal direction corresponding to the largest eigenvalue of the covariance of the vectors that fall into a node. This is equivalent to the PAC Tree [McNames, 2001] with the exception that the splitting threshold (i.e., the β-fractile point) is chosen randomly, rather than setting it to the median point. Sinha [2014] shows that, with the modified algorithm, a smaller ensemble of trees is necessary to reach high retrieval accuracy, as compared with the original RP Tree construction. | 2401.09350#150 | 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": 150,
"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": "52\n4 Branch-and-Bound Algorithms\nNames, 2001], and Max-Margin Trees [Ram et al., 2012] offer other ways of choosing the axis or direction along which the algorithm partitions the data. Vantage-point Trees [Yianilos, 1993], as another example, follow the same iterative procedure as k-d Trees, but partition the space using hyperspheres rather than hyperplanes.\nThere are also various other randomized constructions of tree index struc- tures for NN search. Panigrahy [2008], for instance, construct a standard k-d Tree over the original data points but, during search, perturb the query point. Repeating the perturb-then-search scheme reduces the failure probability of a defeatist search over the k-d Tree.\nSinha [2014] proposes a different variant of the RP Tree where, instead of a random projection, they choose the principal direction corresponding to the largest eigenvalue of the covariance of the vectors that fall into a node. This is equivalent to the PAC Tree [McNames, 2001] with the exception that the splitting threshold (i.e., the β-fractile point) is chosen randomly, rather than setting it to the median point. Sinha [2014] shows that, with the modified algorithm, a smaller ensemble of trees is necessary to reach high retrieval accuracy, as compared with the original RP Tree construction.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 151 | Sinha and Keivani [2017] improve the space complexity of RP Trees by replacing the d-dimensional dense random direction with a sparse random projection using Fast Johnson-Lindenstrauss Transform [Ailon and Chazelle, 2009]. The result is that, every internal node of the tree has to store a sparse vector whose number of non-zero coordinates is far less than d. This space- efficient variant of the RP Tree offers virtually the same theoretical guarantees as the original RP Tree structure.
Ram and Sinha [2019] improve the running time of the NN search over an RP Tree (which is O(d log m) for m = |X |) by first randomly rotating the vectors in a pre-processing step, then applying the standard k-d Tree to the rotated vectors. They show that, such a construction leads to a search time complexity of O(d log d + log m) and offers the same guarantees on the failure probability as the RP Tree. | 2401.09350#151 | 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": 151,
"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": "Sinha and Keivani [2017] improve the space complexity of RP Trees by replacing the d-dimensional dense random direction with a sparse random projection using Fast Johnson-Lindenstrauss Transform [Ailon and Chazelle, 2009]. The result is that, every internal node of the tree has to store a sparse vector whose number of non-zero coordinates is far less than d. This space- efficient variant of the RP Tree offers virtually the same theoretical guarantees as the original RP Tree structure.\nRam and Sinha [2019] improve the running time of the NN search over an RP Tree (which is O(d log m) for m = |X |) by first randomly rotating the vectors in a pre-processing step, then applying the standard k-d Tree to the rotated vectors. They show that, such a construction leads to a search time complexity of O(d log d + log m) and offers the same guarantees on the failure probability as the RP Tree.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 152 | Cover Trees too have been the center of much research. As we have al- ready mentioned, many subsequent works [Elkin and Kurlin, 2022, 2023, Curtin, 2016] investigated the theoretical results presented in the original paper [Beygelzimer et al., 2006] and corrected or improved the time com- plexity bounds on the insertion and search algorithms. Izbicki and Shelton [2015] simplified the structure of the concrete Cover Tree to make its imple- mentation more efficient and cache-aware. Gu et al. [2022] proposed parallel insertion and deletion algorithms for the Cover Tree to scale the algorithm to real-world vector collections. We should also note that the Cover Tree itself is an extension (or, rather, a simplification) of Navigating Nets [Krauthgamer and Lee, 2004], which itself has garnered much research.
It is also possible to extend the framework to MIPS. That may be surpris- ing. After all, the machinery of the branch-and-bound framework rests on the
4.5 Closing Remarks
assumption that the distance function has all the nice properties we expect from a metric space. In particular, we take for granted that the distance is non-negative and that distances obey the triangle inequality. As we know, however, none of these properties holds when the distance function is inner product. | 2401.09350#152 | 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": 152,
"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": "Cover Trees too have been the center of much research. As we have al- ready mentioned, many subsequent works [Elkin and Kurlin, 2022, 2023, Curtin, 2016] investigated the theoretical results presented in the original paper [Beygelzimer et al., 2006] and corrected or improved the time com- plexity bounds on the insertion and search algorithms. Izbicki and Shelton [2015] simplified the structure of the concrete Cover Tree to make its imple- mentation more efficient and cache-aware. Gu et al. [2022] proposed parallel insertion and deletion algorithms for the Cover Tree to scale the algorithm to real-world vector collections. We should also note that the Cover Tree itself is an extension (or, rather, a simplification) of Navigating Nets [Krauthgamer and Lee, 2004], which itself has garnered much research.\nIt is also possible to extend the framework to MIPS. That may be surpris- ing. After all, the machinery of the branch-and-bound framework rests on the\n4.5 Closing Remarks\nassumption that the distance function has all the nice properties we expect from a metric space. In particular, we take for granted that the distance is non-negative and that distances obey the triangle inequality. As we know, however, none of these properties holds when the distance function is inner product.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 153 | As Bachrach et al. [2014] show, however, it is possible to apply a rank- preserving transformation to vectors such that solving MIPS over the original space is equivalent to solving NN over the transformed space. Ram and Gray [2012] take a different approach and derive bounds on the inner product between an arbitrary query point and vectors that are contained in a ball associated with an internal node of the tree index. This bound allows the certification process to proceed as usual. Nonetheless, these methods face the same challenges as k-d Trees and their variants.
# 4.5.2 Future Directions
The literature on branch-and-bound algorithms for top-k retrieval is rather mature and stable at the time of this writing. While publications on this fasci- nating class of algorithms continue to date, most recent works either improve the theoretical analysis of existing algorithms (e.g., [Elkin and Kurlin, 2023]), improve their implementation (e.g., [Ram and Sinha, 2019]), or adapt their implementation to other computing paradigms such as distributed systems (e.g., [Gu et al., 2022]). | 2401.09350#153 | 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": 153,
"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 Bachrach et al. [2014] show, however, it is possible to apply a rank- preserving transformation to vectors such that solving MIPS over the original space is equivalent to solving NN over the transformed space. Ram and Gray [2012] take a different approach and derive bounds on the inner product between an arbitrary query point and vectors that are contained in a ball associated with an internal node of the tree index. This bound allows the certification process to proceed as usual. Nonetheless, these methods face the same challenges as k-d Trees and their variants.\n# 4.5.2 Future Directions\nThe literature on branch-and-bound algorithms for top-k retrieval is rather mature and stable at the time of this writing. While publications on this fasci- nating class of algorithms continue to date, most recent works either improve the theoretical analysis of existing algorithms (e.g., [Elkin and Kurlin, 2023]), improve their implementation (e.g., [Ram and Sinha, 2019]), or adapt their implementation to other computing paradigms such as distributed systems (e.g., [Gu et al., 2022]).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 154 | Indeed, such research is essential. Tree indices areâas the reader will un- doubtedly learn after reading this monographâamong the few retrieval al- gorithms that rest on a sound theoretical foundation. Crucially, their imple- mentations too reflect those theoretical principles: There is little to no gap between theoretical tree indices and their concrete forms. Improving their the- oretical guarantees and modernizing their implementation, therefore, makes a great deal of sense, especially so because works like [Ram and Sinha, 2019] show how competitive tree indices can be in practice. | 2401.09350#154 | 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": 154,
"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": "Indeed, such research is essential. Tree indices areâas the reader will un- doubtedly learn after reading this monographâamong the few retrieval al- gorithms that rest on a sound theoretical foundation. Crucially, their imple- mentations too reflect those theoretical principles: There is little to no gap between theoretical tree indices and their concrete forms. Improving their the- oretical guarantees and modernizing their implementation, therefore, makes a great deal of sense, especially so because works like [Ram and Sinha, 2019] show how competitive tree indices can be in practice.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 155 | An example area that has received little attention concerns the data struc- ture that materializes a tree index. In most works, trees appear in their na¨ıve form and are processed trivially. That is, a tree is simply a collection of if-else blocks, and is evaluated from root to leaf, one node at a time. The vectors in the leaf of a tree, too, are simply searched exhaustively. Importantly, the knowledge that one tree is often insufficient and that a forest of trees is often necessary to reach an acceptable retrieval accuracy, is not taken advantage of. This insight was key in improving forest traversal in the learning-to-rank literature [Lucchese et al., 2015, Ye et al., 2018], in particular when a batch of queries is to be processed simultaneously. It remains to be seen if a more efficient tree traversal algorithm can unlock the power of tree indices.
53
54
4 Branch-and-Bound Algorithms
Perhaps more importantly, the algorithms we studied in this chapter give us an arsenal of theoretical tools that may be of independent interest. The concepts such as partitioning, spillage, and ϵ-nets that are so critical in the development of many of the algorithms we saw earlier, are useful not only in the context of trees, but also in other classes of retrieval algorithms. We will say more on that in future chapters.
# References | 2401.09350#155 | 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": 155,
"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": "An example area that has received little attention concerns the data struc- ture that materializes a tree index. In most works, trees appear in their na¨ıve form and are processed trivially. That is, a tree is simply a collection of if-else blocks, and is evaluated from root to leaf, one node at a time. The vectors in the leaf of a tree, too, are simply searched exhaustively. Importantly, the knowledge that one tree is often insufficient and that a forest of trees is often necessary to reach an acceptable retrieval accuracy, is not taken advantage of. This insight was key in improving forest traversal in the learning-to-rank literature [Lucchese et al., 2015, Ye et al., 2018], in particular when a batch of queries is to be processed simultaneously. It remains to be seen if a more efficient tree traversal algorithm can unlock the power of tree indices.\n53\n54\n4 Branch-and-Bound Algorithms\nPerhaps more importantly, the algorithms we studied in this chapter give us an arsenal of theoretical tools that may be of independent interest. The concepts such as partitioning, spillage, and ϵ-nets that are so critical in the development of many of the algorithms we saw earlier, are useful not only in the context of trees, but also in other classes of retrieval algorithms. We will say more on that in future chapters.\n# References",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 156 | # References
N. Ailon and B. Chazelle. The fast johnsonâlindenstrauss transform and approximate nearest neighbors. SIAM Journal on Computing, 39(1):302â 322, 2009.
Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014.
J. L. Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):509â517, 9 1975.
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.
P. Ciaccia, M. Patella, and P. Zezula. M-tree: An efficient access method for similarity search in metric spaces. In Proceedings of the 23rd International Conference on Very Large Data Bases, page 426â435, 1997. | 2401.09350#156 | 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": 156,
"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": "# References\nN. Ailon and B. Chazelle. The fast johnsonâlindenstrauss transform and approximate nearest neighbors. SIAM Journal on Computing, 39(1):302â 322, 2009.\nY. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014.\nJ. L. Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):509â517, 9 1975.\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.\nP. Ciaccia, M. Patella, and P. Zezula. M-tree: An efficient access method for similarity search in metric spaces. In Proceedings of the 23rd International Conference on Very Large Data Bases, page 426â435, 1997.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 157 | K. L. Clarkson. Nearest neighbor queries in metric spaces. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 609â617, 1997.
R. R. Curtin. Improving dual-tree algorithms. PhD thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2016.
S. Dasgupta and K. Sinha. Randomized partition trees for nearest neighbor search. Algorithmica, 72(1):237â263, 5 2015.
Y. Elkin and V. Kurlin. Counterexamples expose gaps in the proof of time complexity for cover trees introduced in 2006. In 2022 Topological Data Analysis and Visualization, pages 9â17, Los Alamitos, CA, 10 2022.
Y. Elkin and V. Kurlin. A new near-linear time algorithm for k-nearest neighbor search using a compressed cover tree. In Proceedings of the 40th International Conference on Machine Learning, 2023.
J. H. Friedman, J. L. Bentley, and R. A. Finkel. An algorithm for finding best matches in logarithmic expected time. ACM Transactions on Mathematical Software, 3(3):209â226, 9 1977. | 2401.09350#157 | 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": 157,
"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. L. Clarkson. Nearest neighbor queries in metric spaces. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 609â617, 1997.\nR. R. Curtin. Improving dual-tree algorithms. PhD thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2016.\nS. Dasgupta and K. Sinha. Randomized partition trees for nearest neighbor search. Algorithmica, 72(1):237â263, 5 2015.\nY. Elkin and V. Kurlin. Counterexamples expose gaps in the proof of time complexity for cover trees introduced in 2006. In 2022 Topological Data Analysis and Visualization, pages 9â17, Los Alamitos, CA, 10 2022.\nY. Elkin and V. Kurlin. A new near-linear time algorithm for k-nearest neighbor search using a compressed cover tree. In Proceedings of the 40th International Conference on Machine Learning, 2023.\nJ. H. Friedman, J. L. Bentley, and R. A. Finkel. An algorithm for finding best matches in logarithmic expected time. ACM Transactions on Mathematical Software, 3(3):209â226, 9 1977.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 158 | Y. Gu, Z. Napier, Y. Sun, and L. Wang. Parallel cover trees and their ap- plications. In Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures, pages 259â272, 2022.
References
M. Izbicki and C. Shelton. Faster cover trees. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 1162â1170, Lille, France, 07â09 Jul 2015.
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.
R. Krauthgamer and J. R. Lee. Navigating nets: Simple algorithms for prox- imity search. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 798â807, 2004.
T. Liu, A. W. Moore, A. Gray, and K. Yang. An investigation of practical approximate nearest neighbor algorithms. In Proceedings of the 17th In- ternational Conference on Neural Information Processing Systems, pages 825â832, 2004. | 2401.09350#158 | 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": 158,
"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": "Y. Gu, Z. Napier, Y. Sun, and L. Wang. Parallel cover trees and their ap- plications. In Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures, pages 259â272, 2022.\nReferences\nM. Izbicki and C. Shelton. Faster cover trees. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 1162â1170, Lille, France, 07â09 Jul 2015.\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.\nR. Krauthgamer and J. R. Lee. Navigating nets: Simple algorithms for prox- imity search. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 798â807, 2004.\nT. Liu, A. W. Moore, A. Gray, and K. Yang. An investigation of practical approximate nearest neighbor algorithms. In Proceedings of the 17th In- ternational Conference on Neural Information Processing Systems, pages 825â832, 2004.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 159 | C. Lucchese, F. M. Nardini, S. Orlando, R. Perego, N. Tonellotto, and R. Ven- turini. Quickscorer: A fast algorithm to rank documents with additive en- sembles of regression trees. In Proceedings of the 38th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 73â82, 2015.
J. McNames. A fast nearest-neighbor algorithm based on a principal axis search tree. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 23(9):964â976, 2001.
R. Panigrahy. An improved algorithm finding nearest neighbor using kd-trees. In LATIN 2008: Theoretical Informatics, pages 387â398, 2008.
P. Ram and A. G. Gray. Maximum inner-product search using cone trees. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, page 931â939, 2012.
P. Ram and K. Sinha. Revisiting kd-tree for nearest neighbor search. In Pro- ceedings of the 25th ACM SIGKDD International Conference on Knowl- edge Discovery and Data Mining, pages 1378â1388, 2019. | 2401.09350#159 | 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": 159,
"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": "C. Lucchese, F. M. Nardini, S. Orlando, R. Perego, N. Tonellotto, and R. Ven- turini. Quickscorer: A fast algorithm to rank documents with additive en- sembles of regression trees. In Proceedings of the 38th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 73â82, 2015.\nJ. McNames. A fast nearest-neighbor algorithm based on a principal axis search tree. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 23(9):964â976, 2001.\nR. Panigrahy. An improved algorithm finding nearest neighbor using kd-trees. In LATIN 2008: Theoretical Informatics, pages 387â398, 2008.\nP. Ram and A. G. Gray. Maximum inner-product search using cone trees. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, page 931â939, 2012.\nP. Ram and K. Sinha. Revisiting kd-tree for nearest neighbor search. In Pro- ceedings of the 25th ACM SIGKDD International Conference on Knowl- edge Discovery and Data Mining, pages 1378â1388, 2019.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 160 | P. Ram, D. Lee, and A. G. Gray. Nearest-neighbor search on a time bud- get via max-margin trees. In Proceedings of the 2012 SIAM International Conference on Data Mining, pages 1011â1022, 2012.
K. Sinha. Lsh vs randomized partition trees: Which one to use for nearest neighbor search? In Proceedings of the 13th International Conference on Machine Learning and Applications, pages 41â46, 2014.
K. Sinha and O. Keivani. Sparse Randomized Partition Trees for Nearest Neighbor Search. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, volume 54 of Proceedings of Machine Learning Research, pages 681â689, 20â22 Apr 2017.
R. F. Sproull. Refinements to nearest-neighbor searching ink-dimensional trees. Algorithmica, 6(1):579â589, 6 1991.
T. Ye, H. Zhou, W. Y. Zou, B. Gao, and R. Zhang. Rapidscorer: Fast tree ensemble evaluation by maximizing compactness in data level paralleliza55
56
4 Branch-and-Bound Algorithms
tion. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 941â950, 2018. | 2401.09350#160 | 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": 160,
"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. Ram, D. Lee, and A. G. Gray. Nearest-neighbor search on a time bud- get via max-margin trees. In Proceedings of the 2012 SIAM International Conference on Data Mining, pages 1011â1022, 2012.\nK. Sinha. Lsh vs randomized partition trees: Which one to use for nearest neighbor search? In Proceedings of the 13th International Conference on Machine Learning and Applications, pages 41â46, 2014.\nK. Sinha and O. Keivani. Sparse Randomized Partition Trees for Nearest Neighbor Search. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, volume 54 of Proceedings of Machine Learning Research, pages 681â689, 20â22 Apr 2017.\nR. F. Sproull. Refinements to nearest-neighbor searching ink-dimensional trees. Algorithmica, 6(1):579â589, 6 1991.\nT. Ye, H. Zhou, W. Y. Zou, B. Gao, and R. Zhang. Rapidscorer: Fast tree ensemble evaluation by maximizing compactness in data level paralleliza55\n56\n4 Branch-and-Bound Algorithms\ntion. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 941â950, 2018.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 161 | 56
4 Branch-and-Bound Algorithms
tion. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 941â950, 2018.
P. N. Yianilos. Data structures and algorithms for nearest neighbor search In Proceedings of the 4th Annual ACM-SIAM in general metric spaces. Symposium on Discrete Algorithms, pages 311â321, 1993.
Chapter 5 Locality Sensitive Hashing
Abstract In the preceding chapter, we delved into algorithms that inferred the geometrical shape of a collection of vectors and condensed it into a navi- gable structure. In many cases, the algorithms were designed for exact top-k retrieval, but could be modified to provide guarantees on approximate search. This section, instead, explores an entirely different idea that is probabilistic in nature and, as such, is designed specifically for approximate top-k retrieval from the ground up.
# 5.1 Intuition | 2401.09350#161 | 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": 161,
"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": "56\n4 Branch-and-Bound Algorithms\ntion. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 941â950, 2018.\nP. N. Yianilos. Data structures and algorithms for nearest neighbor search In Proceedings of the 4th Annual ACM-SIAM in general metric spaces. Symposium on Discrete Algorithms, pages 311â321, 1993.\nChapter 5 Locality Sensitive Hashing\nAbstract In the preceding chapter, we delved into algorithms that inferred the geometrical shape of a collection of vectors and condensed it into a navi- gable structure. In many cases, the algorithms were designed for exact top-k retrieval, but could be modified to provide guarantees on approximate search. This section, instead, explores an entirely different idea that is probabilistic in nature and, as such, is designed specifically for approximate top-k retrieval from the ground up.\n# 5.1 Intuition",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 162 | # 5.1 Intuition
Let us consider the intuition behind what is known as Locality Sensitive Hashing (LSH) [Indyk and Motwani, 1998] first. Define b separate âbuckets.â Now, suppose there exists a mapping h(·) from vectors in Rd to these buckets, such that every vector is placed into a single bucket: h : Rd â [b]. Crucially, assume that vectors that are closer to each other according to the distance function δ(·, ·), are more likely to be placed into the same bucket. In other words, the probability that two vectors collide increases as δ decreases.
Considering the setup above, indexing is simply a matter of applying h to all vectors in the collection X and making note of the resulting placements. Retrieval for a query q is also straightforward: Perform exact search over the data points that are in the bucket h(q). The reason this procedure works with high probability is because it is more likely for the mapping h to place q in a bucket that contains its nearest neighbors, so that an exact search over the h(q) bucket yields the correct top-k vectors with high likelihood. This is visualized in Figure 5.1(a). | 2401.09350#162 | 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": 162,
"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": "# 5.1 Intuition\nLet us consider the intuition behind what is known as Locality Sensitive Hashing (LSH) [Indyk and Motwani, 1998] first. Define b separate âbuckets.â Now, suppose there exists a mapping h(·) from vectors in Rd to these buckets, such that every vector is placed into a single bucket: h : Rd â [b]. Crucially, assume that vectors that are closer to each other according to the distance function δ(·, ·), are more likely to be placed into the same bucket. In other words, the probability that two vectors collide increases as δ decreases.\nConsidering the setup above, indexing is simply a matter of applying h to all vectors in the collection X and making note of the resulting placements. Retrieval for a query q is also straightforward: Perform exact search over the data points that are in the bucket h(q). The reason this procedure works with high probability is because it is more likely for the mapping h to place q in a bucket that contains its nearest neighbors, so that an exact search over the h(q) bucket yields the correct top-k vectors with high likelihood. This is visualized in Figure 5.1(a).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 163 | It is easy to extend this setup to âmulti-dimensionalâ buckets in the fol- lowing sense. If hiâs are independent functions that have the desired prop- erty above (i.e., increased chance of collision with smaller δ), we may define a bucket in [b]â as the vector mapping g(·) = [h1(·), h2(·), . . . , hâ(·)]. Fig57
58
5 Locality Sensitive Hashing
C7
w (0) Pe SX Oo Yo | \ @/ iO | 8 @e® lo | J ig | / e \ 2| | âup| OS PS 7 HE
(a) (b) | 2401.09350#163 | 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": 163,
"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 easy to extend this setup to âmulti-dimensionalâ buckets in the fol- lowing sense. If hiâs are independent functions that have the desired prop- erty above (i.e., increased chance of collision with smaller δ), we may define a bucket in [b]â as the vector mapping g(·) = [h1(·), h2(·), . . . , hâ(·)]. Fig57\n58\n5 Locality Sensitive Hashing\nC7\nw (0) Pe SX Oo Yo | \\ @/ iO | 8 @e® lo | J ig | / e \\ 2| | âup| OS PS 7 HE\n(a) (b)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 164 | (a) (b)
Fig. 5.1: Illustration of Locality Sensitive Hashing. In (a), a function h : R2 â {1, 2, 3, 4} maps vectors to four buckets. Ideally, when two points are closer to each other, they are more likely to be placed in the same bucket. But, as the dashed arrows show, some vectors end up in less-than-ideal buckets. When retrieving the top-k vectors for a query q, we search through the data vectors that are in the bucket h(q). Figure (b) depicts an extension of the framework where each bucket is the vector [h1(·), h2(·)] obtained from two independent mappings h1 and h2.
ure 5.1(b) illustrates this extension for â = 2. The indexing and search proce- dures work in much the same way. But now, there are presumably fewer data points in each bucket, and spurious collisions (i.e., vectors that were mapped to the same bucket but that are far from each other according to δ) are less likely to occur. In this way, we are likely to reduce the overall search time and increase the accuracy of the algorithm. | 2401.09350#164 | 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": 164,
"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. 5.1: Illustration of Locality Sensitive Hashing. In (a), a function h : R2 â {1, 2, 3, 4} maps vectors to four buckets. Ideally, when two points are closer to each other, they are more likely to be placed in the same bucket. But, as the dashed arrows show, some vectors end up in less-than-ideal buckets. When retrieving the top-k vectors for a query q, we search through the data vectors that are in the bucket h(q). Figure (b) depicts an extension of the framework where each bucket is the vector [h1(·), h2(·)] obtained from two independent mappings h1 and h2.\nure 5.1(b) illustrates this extension for â = 2. The indexing and search proce- dures work in much the same way. But now, there are presumably fewer data points in each bucket, and spurious collisions (i.e., vectors that were mapped to the same bucket but that are far from each other according to δ) are less likely to occur. In this way, we are likely to reduce the overall search time and increase the accuracy of the algorithm.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 165 | Extending the framework even further, we can repeat the process above L times by constructing independent mappings g1(·) through gL(·) from indi- vidual mappings hij(·) (1 ⤠i ⤠L and 1 ⤠j ⤠â), all of which possessing the property of interest. Because the mappings are independent, repeating the procedure many times increases the probability of obtaining a high retrieval accuracy.
That is the essence of the LSH approach to top-k retrieval. Its key ingredi- ent is the family H of functions hijâs that have the stated property for a given distance function, δ. This is the detail that is studied in the remainder of this section. But before we proceed to define H for different distance functions, we will first give a more rigorous description of the algorithm.
5.2 Top-k Retrieval with LSH
# 5.2 Top-k Retrieval with LSH
Earlier, we described informally the class of mappings that are at the core of LSH, as hash functions that preserve the distance between points. That is, the likelihood that such a hash function places two points in the same bucket is a function of their distance. Let us formalize that notion first in the following definition, due to Indyk and Motwani [1998]. | 2401.09350#165 | 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": 165,
"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": "Extending the framework even further, we can repeat the process above L times by constructing independent mappings g1(·) through gL(·) from indi- vidual mappings hij(·) (1 ⤠i ⤠L and 1 ⤠j ⤠â), all of which possessing the property of interest. Because the mappings are independent, repeating the procedure many times increases the probability of obtaining a high retrieval accuracy.\nThat is the essence of the LSH approach to top-k retrieval. Its key ingredi- ent is the family H of functions hijâs that have the stated property for a given distance function, δ. This is the detail that is studied in the remainder of this section. But before we proceed to define H for different distance functions, we will first give a more rigorous description of the algorithm.\n5.2 Top-k Retrieval with LSH\n# 5.2 Top-k Retrieval with LSH\nEarlier, we described informally the class of mappings that are at the core of LSH, as hash functions that preserve the distance between points. That is, the likelihood that such a hash function places two points in the same bucket is a function of their distance. Let us formalize that notion first in the following definition, due to Indyk and Motwani [1998].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 166 | Definition 5.1 ((r, (1 + ϵ)r, p1, p2)-Sensitive Family) A family of hash functions H = {h : Rd â [b]} is called (r, (1 + ϵ)r, p1, p2)-sensitive for a distance function δ(·, ·), where ϵ > 0 and 0 < p1, p2 < 1, if for any two points u, v â Rd: ⢠δ(u, v) ⤠r =â PH ⢠δ(u, v) > (1 + ϵ)r =â PH
© S(u,v) <r => Py [A(u) = h(v)] > pi; and, © S(u,v) >(1+e)r => Px [h(u) = h(v)] < pr. It is clear that such a family is useful only when p; > po. We will see examples of H for different distance functions later in this section. For the time being, however, suppose such a family of functions exists for any 6 of interest. | 2401.09350#166 | 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": 166,
"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": "Definition 5.1 ((r, (1 + ϵ)r, p1, p2)-Sensitive Family) A family of hash functions H = {h : Rd â [b]} is called (r, (1 + ϵ)r, p1, p2)-sensitive for a distance function δ(·, ·), where ϵ > 0 and 0 < p1, p2 < 1, if for any two points u, v â Rd: ⢠δ(u, v) ⤠r =â PH ⢠δ(u, v) > (1 + ϵ)r =â PH\n© S(u,v) <r => Py [A(u) = h(v)] > pi; and, © S(u,v) >(1+e)r => Px [h(u) = h(v)] < pr. It is clear that such a family is useful only when p; > po. We will see examples of H for different distance functions later in this section. For the time being, however, suppose such a family of functions exists for any 6 of interest.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 167 | The indexing algorithm remains as described before. Fix parameters ¢ and L to be determined later in this section. Then define the vector function g() = [hi(), ha(-),---, he(-)] where h; ⬠H. Now, construct L such functions g, through g,, and process the data points in collection Â¥ by evaluating g;âs and placing them in the corresponding multi-dimensional bucket.
In the end, we have effectively built L tables, each mapping buckets to a list of data points that fall into them. Note that, each of the L tables holds a copy of the collection, but where each table organizes the data points differently.
# 5.2.1 The Point Location in Equal Balls Problem
Our intuitive description of retrieval using LSH ignored a minor technicality that we must elaborate in this section. In particular, as is clear from Defi- nition 5.1, a family H has a dependency on the distance r. That means any instance of the family provides guarantees only with respect to a specific r. Consequently, any index obtained from a family H, too, is only useful in the context of a fixed r. | 2401.09350#167 | 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": 167,
"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 indexing algorithm remains as described before. Fix parameters ¢ and L to be determined later in this section. Then define the vector function g() = [hi(), ha(-),---, he(-)] where h; ⬠H. Now, construct L such functions g, through g,, and process the data points in collection Â¥ by evaluating g;âs and placing them in the corresponding multi-dimensional bucket.\nIn the end, we have effectively built L tables, each mapping buckets to a list of data points that fall into them. Note that, each of the L tables holds a copy of the collection, but where each table organizes the data points differently.\n# 5.2.1 The Point Location in Equal Balls Problem\nOur intuitive description of retrieval using LSH ignored a minor technicality that we must elaborate in this section. In particular, as is clear from Defi- nition 5.1, a family H has a dependency on the distance r. That means any instance of the family provides guarantees only with respect to a specific r. Consequently, any index obtained from a family H, too, is only useful in the context of a fixed r.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 168 | It appears, then, that the LSH index is not in and of itself sufficient for solving the ϵ-approximate retrieval problem of Definition 1.2 directly. But, it is enough for solving an easier decision problem that is known as Point Location in Equal Balls (PLEB), defined as follows:
59
60
5 Locality Sensitive Hashing
Definition 5.2 ((r, (1+ϵ)r)-Point Location in Equal Balls) For a query point q and a collection X , if there is a point u â X such that δ(q, u) ⤠r, return Yes and any point v such that δ(q, v) < (1 + ϵ)r. Return No if there are no such points.
The algorithm to solve the (r, (1 + ϵ)r)-PLEB problem for a query point q is fairly straightforward. It involves evaluating giâs on q and exhaustively searching the corresponding buckets in order. We may terminate early af- ter visiting at most 4L data points. For every examined data point u, the algorithm returns Yes if δ(q, u) ⤠(1 + ϵ)r, and No otherwise.
# 5.2.1.1 Proof of Correctness | 2401.09350#168 | 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": 168,
"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 appears, then, that the LSH index is not in and of itself sufficient for solving the ϵ-approximate retrieval problem of Definition 1.2 directly. But, it is enough for solving an easier decision problem that is known as Point Location in Equal Balls (PLEB), defined as follows:\n59\n60\n5 Locality Sensitive Hashing\nDefinition 5.2 ((r, (1+ϵ)r)-Point Location in Equal Balls) For a query point q and a collection X , if there is a point u â X such that δ(q, u) ⤠r, return Yes and any point v such that δ(q, v) < (1 + ϵ)r. Return No if there are no such points.\nThe algorithm to solve the (r, (1 + ϵ)r)-PLEB problem for a query point q is fairly straightforward. It involves evaluating giâs on q and exhaustively searching the corresponding buckets in order. We may terminate early af- ter visiting at most 4L data points. For every examined data point u, the algorithm returns Yes if δ(q, u) ⤠(1 + ϵ)r, and No otherwise.\n# 5.2.1.1 Proof of Correctness",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 169 | # 5.2.1.1 Proof of Correctness
Suppose there exits a point u* ⬠¥ such that 6(g,u*) < r. The algorithm above is correct, in the sense that it returns a point u with d(q,u) < (L+e)r, if we choose ¢ and L such that the following two properties hold with constant probability: e Jie [L] s.t. gi(u*) = gi(q); and, e ya [(*\ Bea (1+e)r yn) 9; Mo, < 4L, where g; '(g,(q)) is the set
e ya [(*\ Bea (1+e)r yn) 9; Mo, < 4L, where g; '(g,(q)) is the set of vectors in bucket g;(q).
The first property ensures that, as we traverse the L buckets associated with the query point, we are likely to visit either the optimal point uâ, or some other point whose distance to q is at most (1 +ϵ)r. The second property guarantees that with constant probability, there are no more than 4L points in the candidate buckets that are (1 + ϵ)r away from q. As such, we are likely to find a solution before visiting 4L points. | 2401.09350#169 | 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": 169,
"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": "# 5.2.1.1 Proof of Correctness\nSuppose there exits a point u* ⬠¥ such that 6(g,u*) < r. The algorithm above is correct, in the sense that it returns a point u with d(q,u) < (L+e)r, if we choose ¢ and L such that the following two properties hold with constant probability: e Jie [L] s.t. gi(u*) = gi(q); and, e ya [(*\\ Bea (1+e)r yn) 9; Mo, < 4L, where g; '(g,(q)) is the set\ne ya [(*\\ Bea (1+e)r yn) 9; Mo, < 4L, where g; '(g,(q)) is the set of vectors in bucket g;(q).\nThe first property ensures that, as we traverse the L buckets associated with the query point, we are likely to visit either the optimal point uâ, or some other point whose distance to q is at most (1 +ϵ)r. The second property guarantees that with constant probability, there are no more than 4L points in the candidate buckets that are (1 + ϵ)r away from q. As such, we are likely to find a solution before visiting 4L points.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 170 | We must therefore prove that for some â and L the above properties hold. The following claim shows one such configuration.
Theorem 5.1 Let Ï = ln p1/ ln p2 and m = |X |. Set L = mÏ and â = log1/p2 m. The properties above hold with constant probability for a (r, (1 + ϵ)r, p1, p2)-sensitive LSH family.
Proof. Consider the first property. We have, from Definition 5.1, that, for any hi â H:
P [hi(u') = hi(q)| > pi.
That holds simply because uâ â B(q, r). That implies:
P [si(u") = ai(a)| > pi
As such:
P [3 i ⬠[L] s.t. gi(u*) = ai(a)| >1-(1- pi).
5.2 Top-k Retrieval with LSH
Substituting â and L with the expressions given in the theorem gives:
: we) Line 1 P[3ie [L] s.t. gi(u )=gila)| 2 1-(Q-7 7) w1- 7,
proving that the property of interest holds with constant probability. | 2401.09350#170 | 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": 170,
"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 must therefore prove that for some â and L the above properties hold. The following claim shows one such configuration.\nTheorem 5.1 Let Ï = ln p1/ ln p2 and m = |X |. Set L = mÏ and â = log1/p2 m. The properties above hold with constant probability for a (r, (1 + ϵ)r, p1, p2)-sensitive LSH family.\nProof. Consider the first property. We have, from Definition 5.1, that, for any hi â H:\nP [hi(u') = hi(q)| > pi.\nThat holds simply because uâ â B(q, r). That implies:\nP [si(u\") = ai(a)| > pi\nAs such:\nP [3 i ⬠[L] s.t. gi(u*) = ai(a)| >1-(1- pi).\n5.2 Top-k Retrieval with LSH\nSubstituting â and L with the expressions given in the theorem gives:\n: we) Line 1 P[3ie [L] s.t. gi(u )=gila)| 2 1-(Q-7 7) w1- 7,\nproving that the property of interest holds with constant probability.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 171 | proving that the property of interest holds with constant probability.
Next, consider the second property. For any point v such that δ(q, v) > (1 + ϵ)r, Definition 5.1 tells us that:
P [hi(v) = hila)] <p2 â> P[gi(v) = gi(a)] < v5 P [gi(w) x(a) <+ m =e v st gi(v) = gig) (4,0) > (1+ Or] [gi] <1 =e v s.t. gi(v) = gi(q) Ad(q,v) > A+ orl] <L,
where the last expression follows by the linearity of expectation when applied to all L buckets. By Markovâs inequality, the probability that there are more than 4L points for which δ(q, v) > (1 + ϵ)r but that map to the same bucket ââ as q is at most 1/4. That completes the proof.
# 5.2.1.2 Space and Time Complexity
The algorithm terminates after visiting at most 4L vectors in the candidate buckets. Given the configuration of Theorem 5.1, this means that the time complexity of the algorithm for query processing is O(dmÏ), which is sub- linear in m. | 2401.09350#171 | 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": 171,
"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": "proving that the property of interest holds with constant probability.\nNext, consider the second property. For any point v such that δ(q, v) > (1 + ϵ)r, Definition 5.1 tells us that:\nP [hi(v) = hila)] <p2 â> P[gi(v) = gi(a)] < v5 P [gi(w) x(a) <+ m =e v st gi(v) = gig) (4,0) > (1+ Or] [gi] <1 =e v s.t. gi(v) = gi(q) Ad(q,v) > A+ orl] <L,\nwhere the last expression follows by the linearity of expectation when applied to all L buckets. By Markovâs inequality, the probability that there are more than 4L points for which δ(q, v) > (1 + ϵ)r but that map to the same bucket ââ as q is at most 1/4. That completes the proof.\n# 5.2.1.2 Space and Time Complexity\nThe algorithm terminates after visiting at most 4L vectors in the candidate buckets. Given the configuration of Theorem 5.1, this means that the time complexity of the algorithm for query processing is O(dmÏ), which is sub- linear in m.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 172 | As for space complexity of the algorithm, note that the index stores each data point L times. That implies the space required to build an LSH index has complexity O(mL) = O(m1+Ï), which grows super-linearly with m. This growth rate can easily become prohibitive [Gionis et al., 1999, Buhler, 2001], particularly because it is often necessary to increase L to reach a higher accuracy, as the proof of Theorem 5.1 shows. How do we reduce this overhead and still obtain sub-linear query time? That is a question that has led to a flurry of research in the past.
One direction to address that question is to modify the search algorithm so that it visits multiple buckets from each of the L tables, instead of examining just a single bucket per table. That is the idea first explored by Panigrahy [2006]. In that work, the search algorithm is the same as in the standard version presented above, but in addition to searching the buckets for query q, it also performs many search operations for perturbed copies of q. While theoretically interesting, their method proves difficult to use in practice. That is because, the amount of noise needed to perturb a query depends on the distance of the nearest neighbor to qâa quantity that is unknown a priori.
61
62
5 Locality Sensitive Hashing | 2401.09350#172 | 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": 172,
"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 for space complexity of the algorithm, note that the index stores each data point L times. That implies the space required to build an LSH index has complexity O(mL) = O(m1+Ï), which grows super-linearly with m. This growth rate can easily become prohibitive [Gionis et al., 1999, Buhler, 2001], particularly because it is often necessary to increase L to reach a higher accuracy, as the proof of Theorem 5.1 shows. How do we reduce this overhead and still obtain sub-linear query time? That is a question that has led to a flurry of research in the past.\nOne direction to address that question is to modify the search algorithm so that it visits multiple buckets from each of the L tables, instead of examining just a single bucket per table. That is the idea first explored by Panigrahy [2006]. In that work, the search algorithm is the same as in the standard version presented above, but in addition to searching the buckets for query q, it also performs many search operations for perturbed copies of q. While theoretically interesting, their method proves difficult to use in practice. That is because, the amount of noise needed to perturb a query depends on the distance of the nearest neighbor to qâa quantity that is unknown a priori.\n61\n62\n5 Locality Sensitive Hashing",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 173 | 61
62
5 Locality Sensitive Hashing
Additionally, it is likely that a single bucket may be visited many times over as we invoke the search procedure on the copies of q.
Later, Lv et al. [2007] refined that theoretical result and presented a method that, instead of perturbing queries randomly and performing mul- tiple hash computations and search invocations, utilizes a more efficient ap- proach in deciding which buckets to probe within each table. In particular, their âmulti-probe LSHâ first finds the bucket associated with q, say gi(q). It then additionally visits other âadjacentâ buckets where a bucket is adjacent if it is more likely to hold data points that are close to the vectors in gi(q). The precise way their algorithm arrives at a set of adjacent buckets de- pends on the hash family itself. In their work, Lv et al. [2007] consider only a hash family for the Euclidean distance, and take advantage of the fact that adjacent buckets (which are in [b]â) differ in each coordinate by at most 1â this becomes clearer when we review the LSH family for Euclidean distance in Section 5.3.3. This scheme was shown empirically to reduce by an order of magnitude the total number of hash tables that is required to achieve an accuracy greater than 0.9 on high-dimensional datasets. | 2401.09350#173 | 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": 173,
"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": "61\n62\n5 Locality Sensitive Hashing\nAdditionally, it is likely that a single bucket may be visited many times over as we invoke the search procedure on the copies of q.\nLater, Lv et al. [2007] refined that theoretical result and presented a method that, instead of perturbing queries randomly and performing mul- tiple hash computations and search invocations, utilizes a more efficient ap- proach in deciding which buckets to probe within each table. In particular, their âmulti-probe LSHâ first finds the bucket associated with q, say gi(q). It then additionally visits other âadjacentâ buckets where a bucket is adjacent if it is more likely to hold data points that are close to the vectors in gi(q). The precise way their algorithm arrives at a set of adjacent buckets de- pends on the hash family itself. In their work, Lv et al. [2007] consider only a hash family for the Euclidean distance, and take advantage of the fact that adjacent buckets (which are in [b]â) differ in each coordinate by at most 1â this becomes clearer when we review the LSH family for Euclidean distance in Section 5.3.3. This scheme was shown empirically to reduce by an order of magnitude the total number of hash tables that is required to achieve an accuracy greater than 0.9 on high-dimensional datasets.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 174 | Another direction is to improve the guarantees of the LSH family itself. As Theorem 5.1 indicates, Ï = log p1/ log p2 plays a critical role in the efficiency and effectiveness of the search algorithm, as well as the space complexity of the data structure. It makes sense, then, that improving Ï leads to smaller space overhead. Many works have explored advanced LSH families to do just that [Andoni and Indyk, 2008, Andoni et al., 2014, 2015]. We review some of these methods in more detail later in this chapter.
# 5.2.2 Back to the Approximate Retrieval Problem
A solution to PLEB of Definition 5.2 is a solution to ϵ-approximate top-k retrieval only if r = δ(q, uâ), where uâ is the k-th minimizer of δ(q, ·). But we do not know the minimal distance in advance! That begs the question: How does solving the PLEB problem help us solve the ϵ-approximate retrieval problem?
Indyk and Motwani [1998] argue that an efficient solution to this decision version of the problem leads directly to an efficient solution to the original problem. In effect, they show that ϵ-approximate retrieval can be reduced to PLEB. Let us review one simple, albeit inefficient reduction. | 2401.09350#174 | 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": 174,
"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": "Another direction is to improve the guarantees of the LSH family itself. As Theorem 5.1 indicates, Ï = log p1/ log p2 plays a critical role in the efficiency and effectiveness of the search algorithm, as well as the space complexity of the data structure. It makes sense, then, that improving Ï leads to smaller space overhead. Many works have explored advanced LSH families to do just that [Andoni and Indyk, 2008, Andoni et al., 2014, 2015]. We review some of these methods in more detail later in this chapter.\n# 5.2.2 Back to the Approximate Retrieval Problem\nA solution to PLEB of Definition 5.2 is a solution to ϵ-approximate top-k retrieval only if r = δ(q, uâ), where uâ is the k-th minimizer of δ(q, ·). But we do not know the minimal distance in advance! That begs the question: How does solving the PLEB problem help us solve the ϵ-approximate retrieval problem?\nIndyk and Motwani [1998] argue that an efficient solution to this decision version of the problem leads directly to an efficient solution to the original problem. In effect, they show that ϵ-approximate retrieval can be reduced to PLEB. Let us review one simple, albeit inefficient reduction.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 175 | Let δâ = maxu,vâX δ(u, v) and δâ = minu,vâX δ(u, v). Denote by â the aspect ratio: â = δâ/δâ. Now, define a set of distances R = {(1 + ϵ)0, (1 + ϵ)1, . . . , â}, and construct |R| LSH indices for each r â R.
Retrieving vectors for query q is a matter of performing binary search over R to find the minimal distance such that PLEB succeeds and returns a point u â X . That point u is the solution to the ϵ-approximate retrieval problem!
5.3 LSH Families
It is easy to see that such a reduction adds to the time complexity by a factor of O(log log1+ϵ â), and to the space complexity by a factor of O(log1+ϵ â).
# 5.3 LSH Families | 2401.09350#175 | 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": 175,
"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 = maxu,vâX δ(u, v) and δâ = minu,vâX δ(u, v). Denote by â the aspect ratio: â = δâ/δâ. Now, define a set of distances R = {(1 + ϵ)0, (1 + ϵ)1, . . . , â}, and construct |R| LSH indices for each r â R.\nRetrieving vectors for query q is a matter of performing binary search over R to find the minimal distance such that PLEB succeeds and returns a point u â X . That point u is the solution to the ϵ-approximate retrieval problem!\n5.3 LSH Families\nIt is easy to see that such a reduction adds to the time complexity by a factor of O(log log1+ϵ â), and to the space complexity by a factor of O(log1+ϵ â).\n# 5.3 LSH Families",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 176 | # 5.3 LSH Families
We have studied how LSH solves the PLEB problem of Definition 5.2, ana- lyzed its time and space complexity, and reviewed how a solution to PLEB leads to a solution to the ϵ-approximate top-k retrieval problem of Defini- tion 1.2. Throughout that discussion, we took for granted the existence of an LSH family that satisfies Definition 5.1 for a distance function of interest. In this section, we review example families and unpack their construction to complete the picture.
# 5.3.1 Hamming Distance
We start with the simpler case of Hamming distance over the space of binary vectors. That is, we assume that X â {0, 1}d and δ(u, v) = â¥u â vâ¥1, measur- ing the number of coordinates in which the two vectors u and v differ. For this setup, a hash family that maps a vector to one of its coordinates at randomâ a technique that is also known as bit samplingâis an LSH family [Indyk and Motwani, 1998], as the claim below shows. | 2401.09350#176 | 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": 176,
"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": "# 5.3 LSH Families\nWe have studied how LSH solves the PLEB problem of Definition 5.2, ana- lyzed its time and space complexity, and reviewed how a solution to PLEB leads to a solution to the ϵ-approximate top-k retrieval problem of Defini- tion 1.2. Throughout that discussion, we took for granted the existence of an LSH family that satisfies Definition 5.1 for a distance function of interest. In this section, we review example families and unpack their construction to complete the picture.\n# 5.3.1 Hamming Distance\nWe start with the simpler case of Hamming distance over the space of binary vectors. That is, we assume that X â {0, 1}d and δ(u, v) = â¥u â vâ¥1, measur- ing the number of coordinates in which the two vectors u and v differ. For this setup, a hash family that maps a vector to one of its coordinates at randomâ a technique that is also known as bit samplingâis an LSH family [Indyk and Motwani, 1998], as the claim below shows.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 177 | Theorem 5.2 For X â {0, 1}d equipped with the Hamming distance, the family H = {hi | hi(u) = ui, 1 ⤠i ⤠d} is (r, (1 + ϵ)r, 1 â r/d, 1 â (1 + ϵ)r/d)- sensitive.
Proof. The proof is trivial. For a given r and two vectors u,v ⬠{0,1}4, if lu â vl]; <r, then P [hi(u) # hi(v)] < r/d, so that P [he(u) = hi(v)] 1âr/d, and therefore pj = 1 â1/d. pz is derived similarly. Oo IV
# 5.3.2 Angular Distance
Consider next the angular distance between two real vectors u, v â Rd, de- fined as:
5(u, v) = arccos (ur), (5.1) Ilel2llella
63
64
5 Locality Sensitive Hashing
©, -)
(a) Hyperplane LSH (b) Cross-polytope LSH | 2401.09350#177 | 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": 177,
"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 5.2 For X â {0, 1}d equipped with the Hamming distance, the family H = {hi | hi(u) = ui, 1 ⤠i ⤠d} is (r, (1 + ϵ)r, 1 â r/d, 1 â (1 + ϵ)r/d)- sensitive.\nProof. The proof is trivial. For a given r and two vectors u,v ⬠{0,1}4, if lu â vl]; <r, then P [hi(u) # hi(v)] < r/d, so that P [he(u) = hi(v)] 1âr/d, and therefore pj = 1 â1/d. pz is derived similarly. Oo IV\n# 5.3.2 Angular Distance\nConsider next the angular distance between two real vectors u, v â Rd, de- fined as:\n5(u, v) = arccos (ur), (5.1) Ilel2llella\n63\n64\n5 Locality Sensitive Hashing\n©, -)\n(a) Hyperplane LSH (b) Cross-polytope LSH",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 178 | Fig. 5.2: Illustration of hyperplane and cross-polytope LSH functions for an- gular distance in R2. In hyperplane LSH, we draw random directions (a and b) to define hyperplanes (A and B), and record +1 or â1 depending on which side of the hyperplane a vector (u and v) lies. For example, ha(u) = â1, ha(v) = +1, and hb(u) = hb(v) = â1. It is easy to see that the probability of a hash collision for two vectors u and v correlates with the angle between them. A cross-polytope LSH function, on the other hand, randomly rotates and normalizes (using matrix A or B) the vector (u), and records the clos- est standard basis vector as its hash. Note that, the cross-polytope is the L1 ball, which in R2 is a rotated square. As an example, hA(u) = âe1 and hB(u) = +e1.
# 5.3.2.1 Hyperplane LSH | 2401.09350#178 | 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": 178,
"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. 5.2: Illustration of hyperplane and cross-polytope LSH functions for an- gular distance in R2. In hyperplane LSH, we draw random directions (a and b) to define hyperplanes (A and B), and record +1 or â1 depending on which side of the hyperplane a vector (u and v) lies. For example, ha(u) = â1, ha(v) = +1, and hb(u) = hb(v) = â1. It is easy to see that the probability of a hash collision for two vectors u and v correlates with the angle between them. A cross-polytope LSH function, on the other hand, randomly rotates and normalizes (using matrix A or B) the vector (u), and records the clos- est standard basis vector as its hash. Note that, the cross-polytope is the L1 ball, which in R2 is a rotated square. As an example, hA(u) = âe1 and hB(u) = +e1.\n# 5.3.2.1 Hyperplane LSH",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 179 | # 5.3.2.1 Hyperplane LSH
For this distance function, one simple LSH family is the set of hash functions that project a vector onto a randomly chosen direction and record the sign of the projection. Put differently, a hash function in this family is characterized by a random hyperplane, which is in turn defined by a unit vector sampled uniformly at random. When applied to an input vector u, the function returns a binary value (from {â1, 1}) indicating on which side of the hyperplane u is located. This procedure, which is known as sign random projections or hy- perplane LSH [Charikar, 2002], is illustrated in Figure 5.2(a) and formalized in the following claim.
Theorem 5.3 For X â Rd equipped with the angular distance of Equa- tion (5.1), the family H = {hr | hr(u) = Sign(â¨r, uâ©), r â¼ Sdâ1} is (θ, (1 + ϵ)θ, 1 â θ/Ï, 1 â (1 + ϵ)θ/Ï)-sensitive for θ â [0, Ï], and Sdâ1 de- noting the d-dimensional hypersphere. | 2401.09350#179 | 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": 179,
"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": "# 5.3.2.1 Hyperplane LSH\nFor this distance function, one simple LSH family is the set of hash functions that project a vector onto a randomly chosen direction and record the sign of the projection. Put differently, a hash function in this family is characterized by a random hyperplane, which is in turn defined by a unit vector sampled uniformly at random. When applied to an input vector u, the function returns a binary value (from {â1, 1}) indicating on which side of the hyperplane u is located. This procedure, which is known as sign random projections or hy- perplane LSH [Charikar, 2002], is illustrated in Figure 5.2(a) and formalized in the following claim.\nTheorem 5.3 For X â Rd equipped with the angular distance of Equa- tion (5.1), the family H = {hr | hr(u) = Sign(â¨r, uâ©), r â¼ Sdâ1} is (θ, (1 + ϵ)θ, 1 â θ/Ï, 1 â (1 + ϵ)θ/Ï)-sensitive for θ â [0, Ï], and Sdâ1 de- noting the d-dimensional hypersphere.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 180 | Proof. If the angle between two vectors is θ, then the probability that a randomly chosen hyperplane lies between them is θ/Ï. As such, the proba5.3 LSH Families
bility that they lie on the same side of the hyperplane is 1 â θ/Ï. The claim ââ follows.
# 5.3.2.2 Cross-polytope LSH
There are a number of other hash families for the angular distance in addition to the basic construction above. Spherical LSH [Andoni et al., 2014] is one example, albeit a purely theoretical oneâa single hash computation from that family alone is considerably more expensive than an exhaustive search over a million data points [Andoni et al., 2015]!
What is known as Cross-polytope LSH [Andoni et al., 2015, Terasawa and Tanaka, 2007] offers similar guarantees as the Spherical LSH but is a more practical construction. A function from this family randomly rotates an input vector first, then outputs the closest signed standard basis vector (eiâs for 1 ⤠i ⤠d) as the hash value. This is illustrated for R2 in Figure 5.2(b), and stated formally in the following result. | 2401.09350#180 | 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": 180,
"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. If the angle between two vectors is θ, then the probability that a randomly chosen hyperplane lies between them is θ/Ï. As such, the proba5.3 LSH Families\nbility that they lie on the same side of the hyperplane is 1 â θ/Ï. The claim ââ follows.\n# 5.3.2.2 Cross-polytope LSH\nThere are a number of other hash families for the angular distance in addition to the basic construction above. Spherical LSH [Andoni et al., 2014] is one example, albeit a purely theoretical oneâa single hash computation from that family alone is considerably more expensive than an exhaustive search over a million data points [Andoni et al., 2015]!\nWhat is known as Cross-polytope LSH [Andoni et al., 2015, Terasawa and Tanaka, 2007] offers similar guarantees as the Spherical LSH but is a more practical construction. A function from this family randomly rotates an input vector first, then outputs the closest signed standard basis vector (eiâs for 1 ⤠i ⤠d) as the hash value. This is illustrated for R2 in Figure 5.2(b), and stated formally in the following result.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 181 | Theorem 5.4 For X â Sdâ1 equipped with the angular distance of Equa- tion (5.1) or equivalently the Euclidean distance, the following family consti- tutes an LSH:
H = {hR | hR(u) = arg min eâ{±ei}d i=1 â¥e â Ru â¥Ruâ¥2 â¥, R â RdÃd, Rij â¼ N (0, 1)},
where N (0, 1) is the standard Gaussian distribution. The probability of colli- sion for unit vectors u, v â Sdâ1 with â¥u â v⥠< Ï is:
2 Tinto = ha(v)| Ss Ind+ O,(Inind).
.
Importantly:
Ï = log p1 log p2 = 1 (1 + ϵ)2 4 â (1 + ϵ)2r2 4 â r2 + o(1). | 2401.09350#181 | 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": 181,
"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 5.4 For X â Sdâ1 equipped with the angular distance of Equa- tion (5.1) or equivalently the Euclidean distance, the following family consti- tutes an LSH:\nH = {hR | hR(u) = arg min eâ{±ei}d i=1 â¥e â Ru â¥Ruâ¥2 â¥, R â RdÃd, Rij â¼ N (0, 1)},\nwhere N (0, 1) is the standard Gaussian distribution. The probability of colli- sion for unit vectors u, v â Sdâ1 with â¥u â v⥠< Ï is:\n2 Tinto = ha(v)| Ss Ind+ O,(Inind).\n.\nImportantly:\nÏ = log p1 log p2 = 1 (1 + ϵ)2 4 â (1 + ϵ)2r2 4 â r2 + o(1).",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 182 | Importantly:
Ï = log p1 log p2 = 1 (1 + ϵ)2 4 â (1 + ϵ)2r2 4 â r2 + o(1).
Proof. We wish to show that, for two unit vectors u, v â Sdâ1 with â¥u â v⥠< Ï , the expression above for the probability of a hash collision is correct. That, indeed, completes the proof of the theorem itself. To show that, we will take advantage of the spherical symmetry of Gaussian random variablesâwe used this property in the proof of Theorem 2.2.
By the spherical symmetry of Gaussians, without loss of generality, we can assume that u = e1, the first standard basis, and v = αe1 + βe2, where α2 + β2 = 1 (so that v has unit norm) and (α â 1)2 + β2 = Ï 2 (because the distance between u and v is Ï ).
Let us now model the collision probability as follows:
65
66
5 Locality Sensitive Hashing
(a) (b)
ax + By = aX1+ BY, ar + By =â(aXi+ Yi)
ME NSG+ 9x) (+eAn | 2401.09350#182 | 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": 182,
"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": "Importantly:\nÏ = log p1 log p2 = 1 (1 + ϵ)2 4 â (1 + ϵ)2r2 4 â r2 + o(1).\nProof. We wish to show that, for two unit vectors u, v â Sdâ1 with â¥u â v⥠< Ï , the expression above for the probability of a hash collision is correct. That, indeed, completes the proof of the theorem itself. To show that, we will take advantage of the spherical symmetry of Gaussian random variablesâwe used this property in the proof of Theorem 2.2.\nBy the spherical symmetry of Gaussians, without loss of generality, we can assume that u = e1, the first standard basis, and v = αe1 + βe2, where α2 + β2 = 1 (so that v has unit norm) and (α â 1)2 + β2 = Ï 2 (because the distance between u and v is Ï ).\nLet us now model the collision probability as follows:\n65\n66\n5 Locality Sensitive Hashing\n(a) (b) \nax + By = aX1+ BY, ar + By =â(aXi+ Yi)\nME NSG+ 9x) (+eAn",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 183 | (a) (b)
ax + By = aX1+ BY, ar + By =â(aXi+ Yi)
ME NSG+ 9x) (+eAn
Fig. 5.3: Illustration of the set SX1,Y1 = {|x| ⤠X1 ⧠|αx + βy| ⤠αX1 + βY1} in (a). Figure (b) visualizes the derivation of Equation (5.5).
P [h(u) h(v)| 2d = 2d P X,Y<N(0,D) = 2d E X1,Â¥i~N (0,1) P [h(u) h(v) ea Vi, [Xi] < MA laX; + BYi| < aX + 6%] d-1 P [IXa] < Xi AlaX2 + BY] < aX + 6Y1] | Ee ( 5. 2)
# P
The first equality is due again to the spherical symmetry of the hash functions and the fact that there are 2d signed standard basis vectors. The second equality simply uses the expressions for u = e1 and v = αe1 + βe2. The final equality follows because of the independence of the coordinates of X and Y , which are sampled from a d-dimensional isotropic Gaussian distribution. | 2401.09350#183 | 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": 183,
"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) \nax + By = aX1+ BY, ar + By =â(aXi+ Yi)\nME NSG+ 9x) (+eAn\nFig. 5.3: Illustration of the set SX1,Y1 = {|x| ⤠X1 ⧠|αx + βy| ⤠αX1 + βY1} in (a). Figure (b) visualizes the derivation of Equation (5.5).\nP [h(u) h(v)| 2d = 2d P X,Y<N(0,D) = 2d E X1,Â¥i~N (0,1) P [h(u) h(v) ea Vi, [Xi] < MA laX; + BYi| < aX + 6%] d-1 P [IXa] < Xi AlaX2 + BY] < aX + 6Y1] | Ee ( 5. 2)\n# P\nThe first equality is due again to the spherical symmetry of the hash functions and the fact that there are 2d signed standard basis vectors. The second equality simply uses the expressions for u = e1 and v = αe1 + βe2. The final equality follows because of the independence of the coordinates of X and Y , which are sampled from a d-dimensional isotropic Gaussian distribution.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 184 | The innermost term in Equation (5.2) is the Gaussian measure of the closed, convex set {|x| ⤠X1 ⧠|αx + βy| ⤠αX1 + βY1}, which is a bounded plane in R2. This set, which we denote by SX1,Y1, is illustrated in Fig- ure 5.3(a). Then we can expand Equation (5.2) as follows:
2d E X1,Yi~N(0,1) x, [Sx] *) 1 =2d P P[S >t |dt. | xaneveou | [Sx..m] 2 | (5.3) 5. (5.4)
We therefore need to expand P[SX1,Y1] in order to complete the expression above. The rest of the proof derives that quantity.
Step 1. Consider P[SX1,Y1 ] = G(SX1,Y1), which is the standard Gaus- sian measure of the set SX1,Y1. In effect, we are interested in G(S) for some bounded convex subset S â R2. We need the following lemma to derive an
5.3 LSH Families | 2401.09350#184 | 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": 184,
"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 innermost term in Equation (5.2) is the Gaussian measure of the closed, convex set {|x| ⤠X1 ⧠|αx + βy| ⤠αX1 + βY1}, which is a bounded plane in R2. This set, which we denote by SX1,Y1, is illustrated in Fig- ure 5.3(a). Then we can expand Equation (5.2) as follows:\n2d E X1,Yi~N(0,1) x, [Sx] *) 1 =2d P P[S >t |dt. | xaneveou | [Sx..m] 2 | (5.3) 5. (5.4)\nWe therefore need to expand P[SX1,Y1] in order to complete the expression above. The rest of the proof derives that quantity.\nStep 1. Consider P[SX1,Y1 ] = G(SX1,Y1), which is the standard Gaus- sian measure of the set SX1,Y1. In effect, we are interested in G(S) for some bounded convex subset S â R2. We need the following lemma to derive an\n5.3 LSH Families",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 185 | 5.3 LSH Families
expression for G(S). But first define µA(r) as the Lebesgue measure of the intersection of a circle of radius r (Sr) with the set A, normalized by the circumference of Sr, so that 0 ⤠µA(r) ⤠1 is a probability measure:
µA(r) â µ(A â© Sr) 2Ïr ,
and denote by âA the distance from the origin to A (i.e., âA â inf{r > 0 | µA(r) > 0}).
# Lemma 5.1 For the closed set A â R2 with µA(r) non-decreasing:
sup (alr) 2) <glay <e-Â¥ r>0
Proof. The upper-bound can be derived as follows:
°° 72/2 ~~ 2 âA?,/2 ata) = | rua(r) Pars [ reâ (dr =e 4a/?, 0 Aa
For the lower-bound:
~ =r? /2 y ear? /2 ry .â (0)? /2 oa) = [ rua(r)-e" / ar > wateâ | reâ / dr = pa(r')e "1 ; 0 fa | 2401.09350#185 | 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": 185,
"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": "5.3 LSH Families\nexpression for G(S). But first define µA(r) as the Lebesgue measure of the intersection of a circle of radius r (Sr) with the set A, normalized by the circumference of Sr, so that 0 ⤠µA(r) ⤠1 is a probability measure:\nµA(r) â µ(A â© Sr) 2Ïr ,\nand denote by âA the distance from the origin to A (i.e., âA â inf{r > 0 | µA(r) > 0}).\n# Lemma 5.1 For the closed set A â R2 with µA(r) non-decreasing:\nsup (alr) 2) <glay <e-Â¥ r>0\nProof. The upper-bound can be derived as follows:\n°° 72/2 ~~ 2 âA?,/2 ata) = | rua(r) Pars [ reâ (dr =e 4a/?, 0 Aa\nFor the lower-bound:\n~ =r? /2 y ear? /2 ry .â (0)? /2 oa) = [ rua(r)-e\" / ar > wateâ | reâ / dr = pa(r')e \"1 ; 0 fa",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 186 | for all rⲠ> 0. The inequality holds because µA(·) is non-decreasing.
Now, K â â SX1,Y1 is a convex set, so for its complement, K â R2, µK(·) is non-decreasing. Using the above lemma, that fact implies the following for small ϵ:
â
ϵ · eâ(1+ϵ)2â2 K /2) ⤠G(K) ⤠eââ2 K /2. â¦(
â
The lower-bound uses the fact that µK (1 + ϵ)âK = â¦( ϵ), because:
HK NS a4eax) = (1+ Ax arecos ( (1+ 0)Anve. (5.5) (a+ a
See Figure 5.3(b) for a helpful illustration.
Since we are interested in the measure of K â = SX1,Y1, we can apply the
result above directly to obtain:
1 â en A(wv)?/2 <P[Sxv]J<1- (ve: et ACua? 2), (5.6)
where we use the notation âK = â(u, v) = min{u, αu + βv}. | 2401.09350#186 | 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": 186,
"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": "for all râ² > 0. The inequality holds because µA(·) is non-decreasing.\nNow, K â â SX1,Y1 is a convex set, so for its complement, K â R2, µK(·) is non-decreasing. Using the above lemma, that fact implies the following for small ϵ:\nâ\nϵ · eâ(1+ϵ)2â2 K /2) ⤠G(K) ⤠eââ2 K /2. â¦(\nâ\nThe lower-bound uses the fact that µK (1 + ϵ)âK = â¦( ϵ), because:\nHK NS a4eax) = (1+ Ax arecos ( (1+ 0)Anve. (5.5) (a+ a\nSee Figure 5.3(b) for a helpful illustration.\nSince we are interested in the measure of K â = SX1,Y1, we can apply the\nresult above directly to obtain:\n1 â en A(wv)?/2 <P[Sxv]J<1- (ve: et ACua? 2), (5.6)\nwhere we use the notation âK = â(u, v) = min{u, αu + βv}.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 187 | where we use the notation âK = â(u, v) = min{u, αu + βv}.
Step 2. For simplicity, first consider the side of Equation (5.6) that does not depend on ϵ, and substitute that into Equation (5.3). We obtain:
67
# Qo
68
5 Locality Sensitive Hashing
1 2a [ P [PISx..v] > emt] dt 0 -X1.Â¥1~N (0,1) 1 = 2a [ P [earn <1- ert|at 0 X1,.Â¥1~N(0,1) ~ = 2 Poon [A0%4,.%) > les (1-2) 2log (1 tm =) Jat. (5.7)
Step 3. We are left with bounding P[â(X1, Y1) ⥠θ]. â(X1, Y1) ⥠θ is, by definition, the set that is the intersection of two half-planes: X1 ⥠θ and αX1 + βY1 ⥠θ. If we denote this set by K, then we are again interested in the Gaussian measure of K. For small ϵ, we can apply the lemma above to show that: | 2401.09350#187 | 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": 187,
"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": "where we use the notation âK = â(u, v) = min{u, αu + βv}.\nStep 2. For simplicity, first consider the side of Equation (5.6) that does not depend on ϵ, and substitute that into Equation (5.3). We obtain:\n67\n# Qo\n68\n5 Locality Sensitive Hashing\n1 2a [ P [PISx..v] > emt] dt 0 -X1.Â¥1~N (0,1) 1 = 2a [ P [earn <1- ert|at 0 X1,.Â¥1~N(0,1) ~ = 2 Poon [A0%4,.%) > les (1-2) 2log (1 tm =) Jat. (5.7)\nStep 3. We are left with bounding P[â(X1, Y1) ⥠θ]. â(X1, Y1) ⥠θ is, by definition, the set that is the intersection of two half-planes: X1 ⥠θ and αX1 + βY1 ⥠θ. If we denote this set by K, then we are again interested in the Gaussian measure of K. For small ϵ, we can apply the lemma above to show that:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 188 | ⦠ϵeâ(1+ϵ)2â2 K ⤠G(K) ⤠eââ2 K /2, (5.8)
where the constant factor in ⦠depends on the angle between the two half- planes. That is because µ(K â© S(1+ϵ)âK ) is ϵ times that angle.
It is easy to see that â2 K = 4 4âÏ 2 · θ2, so that we arrive at the following for small ϵ and every θ ⥠0:
4 ; ; Q, (ce 46) ae A(X, Yi) > 6| <eE =, (59) Je xrbvon! X1,Â¥i~N (0,1)
Step 4. Substituting Equation (5.9) into Equation (5.7) yields:
2a [ x yee.) [A(x Â¥1) 2 4/2 log (1 - tr) dt 1 4 =a | (t= ye" at 0 1 =2d(dâ1) [ G- 2) xt dt 0 8âT = 2d(dâ )B(7âS:d- 1) =2d0,(1)d = 7,
where B denotes the Beta function and the last step uses the Stirling ap- proximation. | 2401.09350#188 | 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": 188,
"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": "⦠ϵeâ(1+ϵ)2â2 K ⤠G(K) ⤠eââ2 K /2, (5.8)\nwhere the constant factor in ⦠depends on the angle between the two half- planes. That is because µ(K â© S(1+ϵ)âK ) is ϵ times that angle.\nIt is easy to see that â2 K = 4 4âÏ 2 · θ2, so that we arrive at the following for small ϵ and every θ ⥠0:\n4 ; ; Q, (ce 46) ae A(X, Yi) > 6| <eE =, (59) Je xrbvon! X1,Â¥i~N (0,1)\nStep 4. Substituting Equation (5.9) into Equation (5.7) yields:\n2a [ x yee.) [A(x Â¥1) 2 4/2 log (1 - tr) dt 1 4 =a | (t= ye\" at 0 1 =2d(dâ1) [ G- 2) xt dt 0 8âT = 2d(dâ )B(7âS:d- 1) =2d0,(1)d = 7,\nwhere B denotes the Beta function and the last step uses the Stirling ap- proximation.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 189 | where B denotes the Beta function and the last step uses the Stirling ap- proximation.
The result above can be expressed as follows:
ln 1 P[h(u) = h(v)] = Ï 2 4 â Ï 2 ln d ± OÏ (1).
Step 5. Repeating Steps 2 through 4 with the expressions that involve ϵ ââ
5.3 LSH Families
Finally, Andoni et al. [2015] show that, instead of applying a random rota- tion using Gaussian random variables, it is sufficient to use a pseudo-random rotation based on Fast Hadamard Transform. In effect, they replace the ran- dom Gaussian matrix R in the construction above with three consecutive applications of HD, where H is the Hadamard matrix and D is a random diagonal sign matrix (where the entries on the diagonal take values from {±1}).
# 5.3.3 Euclidean Distance
Datar et al. [2004] proposed the first LSH family for the Euclidean distance, δ(u, v) = â¥u â vâ¥2. Their construction relies on the notion of p-stable distri- butions which we define first. | 2401.09350#189 | 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": 189,
"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": "where B denotes the Beta function and the last step uses the Stirling ap- proximation.\nThe result above can be expressed as follows:\nln 1 P[h(u) = h(v)] = Ï 2 4 â Ï 2 ln d ± OÏ (1).\nStep 5. Repeating Steps 2 through 4 with the expressions that involve ϵ ââ\n5.3 LSH Families\nFinally, Andoni et al. [2015] show that, instead of applying a random rota- tion using Gaussian random variables, it is sufficient to use a pseudo-random rotation based on Fast Hadamard Transform. In effect, they replace the ran- dom Gaussian matrix R in the construction above with three consecutive applications of HD, where H is the Hadamard matrix and D is a random diagonal sign matrix (where the entries on the diagonal take values from {±1}).\n# 5.3.3 Euclidean Distance\nDatar et al. [2004] proposed the first LSH family for the Euclidean distance, δ(u, v) = â¥u â vâ¥2. Their construction relies on the notion of p-stable distri- butions which we define first.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 190 | Definition 5.3 (p-stable Distribution) A distribution D, is said to be p- stable if ean a;Z;, where a; ⬠R and Z; ~ Dy, has the same distribution as lol|pZ, where a = [a1,a2,...,,] and Z ~ Dy. As an example, the Gaussian distribution is 2-stable.
Let us state this property slightly differently so it is easier to understand its connection to LSH. Suppose we have an arbitrary vector u â Rd. If we con- struct a d-dimensional random vector α whose coordinates are independently sampled from a p-stable distribution Dp, then the inner product â¨Î±, uâ© is dis- tributed according to â¥uâ¥pZ where Z â¼ Dp. By linearity of inner product, we can also see that â¨Î±, uâ© â â¨Î±, vâ©, for two vectors u, v â Rd, is distributed as â¥u â vâ¥pZ. This particular fact plays an important role in the proof of the following result. | 2401.09350#190 | 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": 190,
"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": "Definition 5.3 (p-stable Distribution) A distribution D, is said to be p- stable if ean a;Z;, where a; ⬠R and Z; ~ Dy, has the same distribution as lol|pZ, where a = [a1,a2,...,,] and Z ~ Dy. As an example, the Gaussian distribution is 2-stable.\nLet us state this property slightly differently so it is easier to understand its connection to LSH. Suppose we have an arbitrary vector u â Rd. If we con- struct a d-dimensional random vector α whose coordinates are independently sampled from a p-stable distribution Dp, then the inner product â¨Î±, uâ© is dis- tributed according to â¥uâ¥pZ where Z â¼ Dp. By linearity of inner product, we can also see that â¨Î±, uâ© â â¨Î±, vâ©, for two vectors u, v â Rd, is distributed as â¥u â vâ¥pZ. This particular fact plays an important role in the proof of the following result.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 191 | Theorem 5.5 For X â Rd equipped with the Euclidean distance, a 2-stable distribution D2, and the uniform distribution U over the interval [0, r], the following family is (r, (1 + ϵ)r, p(r), p((1 + ϵ)r))-sensitive:
H = {hα,β | hα,β(u) = â â¨Î±, uâ© + β r â, α â Rd, αi â¼ D2, β â¼ U [0, r]},
where:
veod= [YC jae
and f is the probability density function of the absolute value of D2.
Proof. The key to proving the claim is modeling the probability of a hash collision for two arbitrary vectors u and v: P . That event can be expressed as follows:
69
70
5 Locality Sensitive Hashing
P [#a.a(u) _ haa()| _p [je + 2) _ [oe + 4) r r =P||(a,u-0)| <raA Besta (a, u) + 8 and (a,v) + 8 do not straddle an integer et BO
|. | 2401.09350#191 | 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": 191,
"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 5.5 For X â Rd equipped with the Euclidean distance, a 2-stable distribution D2, and the uniform distribution U over the interval [0, r], the following family is (r, (1 + ϵ)r, p(r), p((1 + ϵ)r))-sensitive:\nH = {hα,β | hα,β(u) = â â¨Î±, uâ© + β r â, α â Rd, αi â¼ D2, β â¼ U [0, r]},\nwhere:\nveod= [YC jae\nand f is the probability density function of the absolute value of D2.\nProof. The key to proving the claim is modeling the probability of a hash collision for two arbitrary vectors u and v: P . That event can be expressed as follows:\n69\n70\n5 Locality Sensitive Hashing\nP [#a.a(u) _ haa()| _p [je + 2) _ [oe + 4) r r =P||(a,u-0)| <raA Besta (a, u) + 8 and (a,v) + 8 do not straddle an integer et BO\n|.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 192 | |.
Using the 2-stability of α, Event A is equivalent to â¥u â vâ¥2|Z| < r, where Z is drawn from D2. The probability of the complement of Event B is simply the ratio between â¨Î±, u â vâ© and r. Putting all that together, we obtain that:
P [ha,a(u) = he,alv)| = re f(A - sea whe) ge 2=0 r " 1 t t i fl (1 at, t=0 |luâ vo" â|luâ Ilo r
where we derived the last equality by the variable change t = zâ¥u â vâ¥2. Therefore, if â¥u â v⥠⤠x:
P [ha,s(u) = ha,s(v)] >/ =A (1- âYat = pix). t=0 ©
It is easy to complete the proof from here.
# 5.3.4 Inner Product
Many of the arguments that establish the existence of an LHS family for a distance function of interest rely on triangle inequality. Inner product as a measure of similarity, however, does not enjoy that property. As such, devel- oping an LSH family for inner product requires that we somehow transform the problem from MIPS to NN search or MCS search, as was the case in Chapter 4. | 2401.09350#192 | 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": 192,
"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": "|.\nUsing the 2-stability of α, Event A is equivalent to â¥u â vâ¥2|Z| < r, where Z is drawn from D2. The probability of the complement of Event B is simply the ratio between â¨Î±, u â vâ© and r. Putting all that together, we obtain that:\nP [ha,a(u) = he,alv)| = re f(A - sea whe) ge 2=0 r \" 1 t t i fl (1 at, t=0 |luâ vo\" â|luâ Ilo r\nwhere we derived the last equality by the variable change t = zâ¥u â vâ¥2. Therefore, if â¥u â v⥠⤠x:\nP [ha,s(u) = ha,s(v)] >/ =A (1- âYat = pix). t=0 ©\nIt is easy to complete the proof from here.\n# 5.3.4 Inner Product\nMany of the arguments that establish the existence of an LHS family for a distance function of interest rely on triangle inequality. Inner product as a measure of similarity, however, does not enjoy that property. As such, devel- oping an LSH family for inner product requires that we somehow transform the problem from MIPS to NN search or MCS search, as was the case in Chapter 4.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 193 | Finding the right transformation that results in improved hash qualityâas determined by Ïâis the question that has been explored by several works in the past [Neyshabur and Srebro, 2015, Shrivastava and Li, 2015, 2014, Yan et al., 2018].
Let us present a simple example. Note that, we may safely assume that queries are unit vectors (i.e., q â Sdâ1), because the norm of the query does not change the outcome of MIPS.
Now, define the transformation ¢g : R¢ + R¢*", first considered by Bachrach et al. [2014], as follows: ga(u) = [u, \/1 â |lul|3]. Apply this transformation to data points in Â¥. Clearly, ||¢a(u)||2 = 1 for all u ⬠4X. Separately, pad the query points with a single 0: ¢,(v) = [v;0] ⬠R41.
ââ
References | 2401.09350#193 | 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": 193,
"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": "Finding the right transformation that results in improved hash qualityâas determined by Ïâis the question that has been explored by several works in the past [Neyshabur and Srebro, 2015, Shrivastava and Li, 2015, 2014, Yan et al., 2018].\nLet us present a simple example. Note that, we may safely assume that queries are unit vectors (i.e., q â Sdâ1), because the norm of the query does not change the outcome of MIPS.\nNow, define the transformation ¢g : R¢ + R¢*\", first considered by Bachrach et al. [2014], as follows: ga(u) = [u, \\/1 â |lul|3]. Apply this transformation to data points in Â¥. Clearly, ||¢a(u)||2 = 1 for all u ⬠4X. Separately, pad the query points with a single 0: ¢,(v) = [v;0] ⬠R41.\nââ\nReferences",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 194 | ââ
References
We can immediately verify that â¨q, uâ© = â¨Ïq(q), Ïd(u)â© for a query q and data point u. But by applying the transformations Ïd(·) and Ïq(·), we have reduced the problem to MCS! As such, we may use any of existing LSH families that we have seen for angular distance in Section 5.3.2 for MIPS.
There has been much debate over the suitability of the standard LSH framework for inner product, with some works extending the framework to what is known as asymmetric LSH [Shrivastava and Li, 2014, 2015]. It turns out, however, that none of that is necessary. In fact, as Neyshabur and Srebro [2015] argued formally and demonstrated empirically, the simple scheme we described above sufficiently addresses MIPS.
# 5.4 Closing Remarks
Much like branch-and-bound algorithms, an LSH approach to top-k retrieval rests on a solid theoretical foundation. There is a direct link between all that is developed theoretically and the accuracy of an LSH-based top-k retrieval system. | 2401.09350#194 | 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": 194,
"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": "ââ\nReferences\nWe can immediately verify that â¨q, uâ© = â¨Ïq(q), Ïd(u)â© for a query q and data point u. But by applying the transformations Ïd(·) and Ïq(·), we have reduced the problem to MCS! As such, we may use any of existing LSH families that we have seen for angular distance in Section 5.3.2 for MIPS.\nThere has been much debate over the suitability of the standard LSH framework for inner product, with some works extending the framework to what is known as asymmetric LSH [Shrivastava and Li, 2014, 2015]. It turns out, however, that none of that is necessary. In fact, as Neyshabur and Srebro [2015] argued formally and demonstrated empirically, the simple scheme we described above sufficiently addresses MIPS.\n# 5.4 Closing Remarks\nMuch like branch-and-bound algorithms, an LSH approach to top-k retrieval rests on a solid theoretical foundation. There is a direct link between all that is developed theoretically and the accuracy of an LSH-based top-k retrieval system.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 195 | Like tree indices, too, the LSH literature is arguably mature. There is therefore not a great deal of open questions left to investigate in its founda- tion, with many recent works instead exploring learnt hash functions or its applications in other domains.
What remains open and exciting in the context of top-k retrieval, however, is the possibility of extending the theory of LSH to explain the success of other retrieval algorithms. We will return to this discussion in Chapter 7.
# References
A. Andoni and P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Communications of the ACM, 51(1): 117â122, 1 2008.
A. Andoni, P. Indyk, H. L. Nguyen, and I. Razenshteyn. Beyond locality- sensitive hashing. In Proceedings of the 2014 Annual ACM-SIAM Sympo- sium on Discrete Algorithms, pages 1018â1028, 2014.
A. Andoni, P. Indyk, T. Laarhoven, I. Razenshteyn, and L. Schmidt. Practical and optimal lsh for angular distance. In Proceedings of the 28th Interna- tional Conference on Neural Information Processing Systems - Volume 1, pages 1225â1233, 2015. | 2401.09350#195 | 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": 195,
"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": "Like tree indices, too, the LSH literature is arguably mature. There is therefore not a great deal of open questions left to investigate in its founda- tion, with many recent works instead exploring learnt hash functions or its applications in other domains.\nWhat remains open and exciting in the context of top-k retrieval, however, is the possibility of extending the theory of LSH to explain the success of other retrieval algorithms. We will return to this discussion in Chapter 7.\n# References\nA. Andoni and P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Communications of the ACM, 51(1): 117â122, 1 2008.\nA. Andoni, P. Indyk, H. L. Nguyen, and I. Razenshteyn. Beyond locality- sensitive hashing. In Proceedings of the 2014 Annual ACM-SIAM Sympo- sium on Discrete Algorithms, pages 1018â1028, 2014.\nA. Andoni, P. Indyk, T. Laarhoven, I. Razenshteyn, and L. Schmidt. Practical and optimal lsh for angular distance. In Proceedings of the 28th Interna- tional Conference on Neural Information Processing Systems - Volume 1, pages 1225â1233, 2015.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 196 | Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014.
71
72
5 Locality Sensitive Hashing
J. Buhler. Efficient large-scale sequence comparison by locality-sensitive hashing. Bioinformatics, 17(5):419â428, 05 2001.
M. S. Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, pages 380â388, 2002.
M. Datar, N. Immorlica, P. Indyk, and V. S. Mirrokni. Locality-sensitive hashing scheme based on p-stable distributions. In Proceedings of the 20th Annual Symposium on Computational Geometry, pages 253â262, 2004. A. Gionis, P. Indyk, and R. Motwani. Similarity search in high dimensions via hashing. In Proceedings of the 25th International Conference on Very Large Data Bases, pages 518â529, 1999. | 2401.09350#196 | 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": 196,
"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": "Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014.\n71\n72\n5 Locality Sensitive Hashing\nJ. Buhler. Efficient large-scale sequence comparison by locality-sensitive hashing. Bioinformatics, 17(5):419â428, 05 2001.\nM. S. Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, pages 380â388, 2002.\nM. Datar, N. Immorlica, P. Indyk, and V. S. Mirrokni. Locality-sensitive hashing scheme based on p-stable distributions. In Proceedings of the 20th Annual Symposium on Computational Geometry, pages 253â262, 2004. A. Gionis, P. Indyk, and R. Motwani. Similarity search in high dimensions via hashing. In Proceedings of the 25th International Conference on Very Large Data Bases, pages 518â529, 1999.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 197 | P. Indyk and R. Motwani. Approximate nearest neighbors: Towards remov- ing the curse of dimensionality. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 604â613, 1998.
Q. Lv, W. Josephson, Z. Wang, M. Charikar, and K. Li. Multi-probe lsh: Efficient indexing for high-dimensional similarity search. In Proceedings of the 33rd International Conference on Very Large Data Bases, pages 950â 961, 2007.
B. Neyshabur and N. Srebro. On symmetric and asymmetric lshs for inner product search. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, pages 1926â 1934, 2015.
R. Panigrahy. Entropy based nearest neighbor search in high dimensions. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pages 1186â1195, 2006.
A. Shrivastava and P. Li. Asymmetric lsh (alsh) for sublinear time maxi- mum inner product search (mips). In Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2, pages 2321â2329, 2014. | 2401.09350#197 | 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": 197,
"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. Indyk and R. Motwani. Approximate nearest neighbors: Towards remov- ing the curse of dimensionality. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 604â613, 1998.\nQ. Lv, W. Josephson, Z. Wang, M. Charikar, and K. Li. Multi-probe lsh: Efficient indexing for high-dimensional similarity search. In Proceedings of the 33rd International Conference on Very Large Data Bases, pages 950â 961, 2007.\nB. Neyshabur and N. Srebro. On symmetric and asymmetric lshs for inner product search. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, pages 1926â 1934, 2015.\nR. Panigrahy. Entropy based nearest neighbor search in high dimensions. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pages 1186â1195, 2006.\nA. Shrivastava and P. Li. Asymmetric lsh (alsh) for sublinear time maxi- mum inner product search (mips). In Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2, pages 2321â2329, 2014.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 198 | Improved asymmetric locality sensitive hashing (alsh) for maximum inner product search (mips). In Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence, pages 812â821, 2015.
K. Terasawa and Y. Tanaka. Spherical lsh for approximate nearest neigh- bor search on unit hypersphere. In Proceedings of the 10th International Conference on Algorithms and Data Structures, pages 27â38, 2007.
X. Yan, J. Li, X. Dai, H. Chen, and J. Cheng. Norm-ranging lsh for maximum inner product search. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 2956â2965, 2018.
# Chapter 6 Graph Algorithms
Abstract We have seen two major classes of algorithms that approach the top-k retrieval problem in their own unique ways. One recursively partitions a vector collection to model its geometry, and the other hashes the vectors into predefined buckets to reduce the search space. Our next class of algorithms takes yet a different view of the question. At a high level, our third approach is to âwalkâ through a collection, hopping from one vector to another, where every hop gets us spatially closer to the optimal solution. This chapter reviews algorithms that use a graph data structure to implement that idea.
# 6.1 Intuition | 2401.09350#198 | 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": 198,
"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": "Improved asymmetric locality sensitive hashing (alsh) for maximum inner product search (mips). In Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence, pages 812â821, 2015.\nK. Terasawa and Y. Tanaka. Spherical lsh for approximate nearest neigh- bor search on unit hypersphere. In Proceedings of the 10th International Conference on Algorithms and Data Structures, pages 27â38, 2007.\nX. Yan, J. Li, X. Dai, H. Chen, and J. Cheng. Norm-ranging lsh for maximum inner product search. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 2956â2965, 2018.\n# Chapter 6 Graph Algorithms\nAbstract We have seen two major classes of algorithms that approach the top-k retrieval problem in their own unique ways. One recursively partitions a vector collection to model its geometry, and the other hashes the vectors into predefined buckets to reduce the search space. Our next class of algorithms takes yet a different view of the question. At a high level, our third approach is to âwalkâ through a collection, hopping from one vector to another, where every hop gets us spatially closer to the optimal solution. This chapter reviews algorithms that use a graph data structure to implement that idea.\n# 6.1 Intuition",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 199 | # 6.1 Intuition
The most natural way to understand a spatial walk through a collection of vectors is by casting it as traversing a (directed) connected graph. As we will see, whether the graph is directed or not depends on the specific algorithm itself. But the graph must regardless be connected, so that there always exists at least one path between every pair of nodes. This ensures that we can walk through the graph no matter where we begin our traversal.
Let us write G(V, E) to refer to such a graph, whose set of vertices or nodes are denoted by V, and its set of edges by E. So, for u, v â V in a directed graph, if (u, v) â E, we may freely move from node u to node v. Hopping from v to u is not possible if (v, u) /â E. Because we often need to talk about the set of nodes that can be reached by a single hop from a node uâknown as the neighbors of uâwe give it a special symbol and define that set as follows: N (u) = {v | (u, v) â E}. | 2401.09350#199 | 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": 199,
"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": "# 6.1 Intuition\nThe most natural way to understand a spatial walk through a collection of vectors is by casting it as traversing a (directed) connected graph. As we will see, whether the graph is directed or not depends on the specific algorithm itself. But the graph must regardless be connected, so that there always exists at least one path between every pair of nodes. This ensures that we can walk through the graph no matter where we begin our traversal.\nLet us write G(V, E) to refer to such a graph, whose set of vertices or nodes are denoted by V, and its set of edges by E. So, for u, v â V in a directed graph, if (u, v) â E, we may freely move from node u to node v. Hopping from v to u is not possible if (v, u) /â E. Because we often need to talk about the set of nodes that can be reached by a single hop from a node uâknown as the neighbors of uâwe give it a special symbol and define that set as follows: N (u) = {v | (u, v) â E}.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 200 | The idea behind the algorithms in this chapter is to construct a graph in the pre-processing phase and use that as an index of a vector collection for top-k retrieval. To do that, we must decide what is a node in the graph (i.e., define the set V), how nodes are linked to each other (E), and, importantly, what the search algorithm looks like.
73
74
6 Graph Algorithms
Solution Gury
Fig. 6.1: Illustration of the greedy traversal algorithm for finding the top-1 solution on an example (undirected) graph. The procedure enters the graph from an arbitrary âentryâ node. It then compares the distance of the node to query q with the distance of its neighbors to q, and either terminates if no neighbor is closer to q than the node itself, or advances to the closest neighbor. It repeats this procedure until the terminal condition is met. The research question in this chapter concerns the construction of the edge set: How do we construct a sparse graph which can be traversed greedily while providing guarantees on the (near-)optimality of the solution
The set of nodes V is easy to construct: Simply designate every vector in the collection X as a unique node in G, so that |X | = |V|. There should, therefore, be no ambiguity if we referred to a node as a vector. We use both terms interchangeably. | 2401.09350#200 | 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": 200,
"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 idea behind the algorithms in this chapter is to construct a graph in the pre-processing phase and use that as an index of a vector collection for top-k retrieval. To do that, we must decide what is a node in the graph (i.e., define the set V), how nodes are linked to each other (E), and, importantly, what the search algorithm looks like.\n73\n74\n6 Graph Algorithms\nSolution Gury\nFig. 6.1: Illustration of the greedy traversal algorithm for finding the top-1 solution on an example (undirected) graph. The procedure enters the graph from an arbitrary âentryâ node. It then compares the distance of the node to query q with the distance of its neighbors to q, and either terminates if no neighbor is closer to q than the node itself, or advances to the closest neighbor. It repeats this procedure until the terminal condition is met. The research question in this chapter concerns the construction of the edge set: How do we construct a sparse graph which can be traversed greedily while providing guarantees on the (near-)optimality of the solution\nThe set of nodes V is easy to construct: Simply designate every vector in the collection X as a unique node in G, so that |X | = |V|. There should, therefore, be no ambiguity if we referred to a node as a vector. We use both terms interchangeably.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 201 | What properties should the edge set E have? To get a sense of what is required of the edge set, it would help to consider the search algorithm first. Suppose we are searching for the top-1 vector closest to query q, and assume that we are, at the moment, at an arbitrary node u in G.
From node u, we can have a look around and assess if any of our neighbors in N (u) is closer to q. By doing so, we find ourselves in one of two situations. Either we encounter no such neighbor, so that u has the smallest distance to q among its neighbors. If that happens, ideally, we want u to also have the smallest distance to q among all vectors. In other words, in an ideal graph, a local optimum coincides with the global optimum.
Alternatively, we may find one such neighbor v â N (u) for which δ(q, v) < δ(q, u) and v = arg minwâN (u) δ(q, w). In that case, the ideal graph is one where the following event takes place: If we moved from u to v, and repeated the process above in the context of N (v) and so on, we will ultimately arrive at a local optimum (which, by the previous condition, is the global optimum). Terminating the algorithm then would therefore give us the optimal solution to the top-1 retrieval problem. | 2401.09350#201 | 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": 201,
"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 properties should the edge set E have? To get a sense of what is required of the edge set, it would help to consider the search algorithm first. Suppose we are searching for the top-1 vector closest to query q, and assume that we are, at the moment, at an arbitrary node u in G.\nFrom node u, we can have a look around and assess if any of our neighbors in N (u) is closer to q. By doing so, we find ourselves in one of two situations. Either we encounter no such neighbor, so that u has the smallest distance to q among its neighbors. If that happens, ideally, we want u to also have the smallest distance to q among all vectors. In other words, in an ideal graph, a local optimum coincides with the global optimum.\nAlternatively, we may find one such neighbor v â N (u) for which δ(q, v) < δ(q, u) and v = arg minwâN (u) δ(q, w). In that case, the ideal graph is one where the following event takes place: If we moved from u to v, and repeated the process above in the context of N (v) and so on, we will ultimately arrive at a local optimum (which, by the previous condition, is the global optimum). Terminating the algorithm then would therefore give us the optimal solution to the top-1 retrieval problem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 202 | Put differently, in an ideal graph, if moving from a node to any of its neighbors does not get us spatially closer to q, it is because the current node is the optimal solution to the top-1 retrieval problem for q.
6.1 Intuition
Algorithm 3: Greedy search algorithm for top-k retrieval over a graph index.
Input: Graph G = (V, E) over collection X with distance δ(·, ·); query point q; entry node s â V; retrieval depth k. Result: Exact top-k solution for q. 1: Q â {s} ; â· Q is a priority queue 2: while Q changed in the previous iteration do 3: 4: 5: 6: 7: 8: 9: end while 10: return Q
On a graph with that property, the procedure of starting from any node in the graph, hopping to a neighbor that is closer to q, and repeating this procedure until no such neighbor exists, gives the optimal solution. That procedure is the familiar best-first-search algorithm, which we illustrate on a toy graph in Figure 6.1. That will be our base search algorithm for top-1 retrieval. | 2401.09350#202 | 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": 202,
"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": "Put differently, in an ideal graph, if moving from a node to any of its neighbors does not get us spatially closer to q, it is because the current node is the optimal solution to the top-1 retrieval problem for q.\n6.1 Intuition\nAlgorithm 3: Greedy search algorithm for top-k retrieval over a graph index.\nInput: Graph G = (V, E) over collection X with distance δ(·, ·); query point q; entry node s â V; retrieval depth k. Result: Exact top-k solution for q. 1: Q â {s} ; â· Q is a priority queue 2: while Q changed in the previous iteration do 3: 4: 5: 6: 7: 8: 9: end while 10: return Q\nOn a graph with that property, the procedure of starting from any node in the graph, hopping to a neighbor that is closer to q, and repeating this procedure until no such neighbor exists, gives the optimal solution. That procedure is the familiar best-first-search algorithm, which we illustrate on a toy graph in Figure 6.1. That will be our base search algorithm for top-1 retrieval.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 203 | Extending the search algorithm to top-k requires a minor modification to the procedure above. It begins by initializing a priority queue of size k. When we visit a new node, we add it to the queue if its distance with q is smaller than the minimum distance among the nodes already in the queue. We keep moving from a node in the queue to its neighbors until the queue stabilizes (i.e., no unseen neighbor of any of the nodes in the queue has a smaller distance to q). This is described in Algorithm 3.
Note that, assuming δ(·, ·) is proper, it is easy to see that the top-1 optimal- ity guarantee immediately implies top-k optimalityâyou should verify this claim as an exercise. It therefore suffices to state our requirements in terms of top-1 optimality alone. So, ideally, E should guarantee that traversing G in a best-first-search manner yields the optimal top-1 solution.
# 6.1.1 The Research Question
It is trivial to construct an edge set that provides the desired optimality guarantee: Simply add an edge between every pair of nodes, completing the graph! The greedy search algorithm described above will take us to the opti- mal solution.
However, such a graph not only has high space complexity, but it also has a linear query time complexity. That is because, the very first step (which
75
76
6 Graph Algorithms | 2401.09350#203 | 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": 203,
"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": "Extending the search algorithm to top-k requires a minor modification to the procedure above. It begins by initializing a priority queue of size k. When we visit a new node, we add it to the queue if its distance with q is smaller than the minimum distance among the nodes already in the queue. We keep moving from a node in the queue to its neighbors until the queue stabilizes (i.e., no unseen neighbor of any of the nodes in the queue has a smaller distance to q). This is described in Algorithm 3.\nNote that, assuming δ(·, ·) is proper, it is easy to see that the top-1 optimal- ity guarantee immediately implies top-k optimalityâyou should verify this claim as an exercise. It therefore suffices to state our requirements in terms of top-1 optimality alone. So, ideally, E should guarantee that traversing G in a best-first-search manner yields the optimal top-1 solution.\n# 6.1.1 The Research Question\nIt is trivial to construct an edge set that provides the desired optimality guarantee: Simply add an edge between every pair of nodes, completing the graph! The greedy search algorithm described above will take us to the opti- mal solution.\nHowever, such a graph not only has high space complexity, but it also has a linear query time complexity. That is because, the very first step (which\n75\n76\n6 Graph Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 204 | However, such a graph not only has high space complexity, but it also has a linear query time complexity. That is because, the very first step (which
75
76
6 Graph Algorithms
also happens to be the last step) involves comparing the distance of q to the entry node, with the distance of q to every other node in the graph! We are better off exhaustively scanning the entire collection in a flat index.
The research question that prompted the algorithms we are about to study in this chapter is whether there exists a relatively sparse graph that has the optimality guarantee we seek or that can instead provide guarantees for the more relaxed, ϵ-approximate top-k retrieval problem.
As we will learn shortly, with a few notable exceptions, all constructions of E proposed thus far in the literature for high-dimensional vectors amount to heuristics that attempt to approximate a theoretical graph but come with no guarantees. In fact, in almost all cases, their worst-case complexity is no better than exhaustive search. Despite that, many of these heuristics work remarkably well in practice on real datasets, making graph-based methods one of the most widely adopted solutions to the approximate top-k retrieval problem. | 2401.09350#204 | 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": 204,
"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": "However, such a graph not only has high space complexity, but it also has a linear query time complexity. That is because, the very first step (which\n75\n76\n6 Graph Algorithms\nalso happens to be the last step) involves comparing the distance of q to the entry node, with the distance of q to every other node in the graph! We are better off exhaustively scanning the entire collection in a flat index.\nThe research question that prompted the algorithms we are about to study in this chapter is whether there exists a relatively sparse graph that has the optimality guarantee we seek or that can instead provide guarantees for the more relaxed, ϵ-approximate top-k retrieval problem.\nAs we will learn shortly, with a few notable exceptions, all constructions of E proposed thus far in the literature for high-dimensional vectors amount to heuristics that attempt to approximate a theoretical graph but come with no guarantees. In fact, in almost all cases, their worst-case complexity is no better than exhaustive search. Despite that, many of these heuristics work remarkably well in practice on real datasets, making graph-based methods one of the most widely adopted solutions to the approximate top-k retrieval problem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 205 | In the remainder of this chapter, we will see classes of theoretical graphs that were developed in adjacent scientific disciplines, but that are seemingly suitable for the (approximate) top-k retrieval problem. As we introduce these graphs, we also examine representative algorithms that aim to build an ap- proximation of such graphs in high dimensions, and review their properties. We note, however, that the literature on graph-based methods is vast and growing still. There is a plethora of studies that experiment with (minor or major) adjustments to the basic idea described earlier, or that empirically compare and contrast different algorithmic flavors on real-world datasets. This chapter does not claim to, nor does it intend to cover the explosion of material on graph-based algorithms. Instead, it limits its scope to the founda- tional principles and ground-breaking works that are theoretically somewhat interesting. We refer the reader to existing reports and surveys for the full spectrum of works on this topic [Wang et al., 2021, Li et al., 2020].
# 6.2 The Delaunay Graph | 2401.09350#205 | 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": 205,
"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 remainder of this chapter, we will see classes of theoretical graphs that were developed in adjacent scientific disciplines, but that are seemingly suitable for the (approximate) top-k retrieval problem. As we introduce these graphs, we also examine representative algorithms that aim to build an ap- proximation of such graphs in high dimensions, and review their properties. We note, however, that the literature on graph-based methods is vast and growing still. There is a plethora of studies that experiment with (minor or major) adjustments to the basic idea described earlier, or that empirically compare and contrast different algorithmic flavors on real-world datasets. This chapter does not claim to, nor does it intend to cover the explosion of material on graph-based algorithms. Instead, it limits its scope to the founda- tional principles and ground-breaking works that are theoretically somewhat interesting. We refer the reader to existing reports and surveys for the full spectrum of works on this topic [Wang et al., 2021, Li et al., 2020].\n# 6.2 The Delaunay Graph",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 206 | # 6.2 The Delaunay Graph
One classical graph that satisfies the conditions we seek and guarantees the optimality of the solution obtained by best-first-search traversal is the De- launay graph [Delaunay, 1934, Fortune, 1997]. It is easier to understand the construction of the Delaunay graph if we consider instead its dual: the Voronoi diagram. So we begin with a description of the Voronoi diagram and Voronoi regions.
6.2 The Delaunay Graph
(a) Voronoi diagram (b) Delaunay graph | 2401.09350#206 | 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": 206,
"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": "# 6.2 The Delaunay Graph\nOne classical graph that satisfies the conditions we seek and guarantees the optimality of the solution obtained by best-first-search traversal is the De- launay graph [Delaunay, 1934, Fortune, 1997]. It is easier to understand the construction of the Delaunay graph if we consider instead its dual: the Voronoi diagram. So we begin with a description of the Voronoi diagram and Voronoi regions.\n6.2 The Delaunay Graph\n(a) Voronoi diagram (b) Delaunay graph",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 207 | Fig. 6.2: Visualization of the Voronoi diagram (a) and its dual, the Delaunay graph (b) for an example collection X of points in R2. A Voronoi region associated with a point u (shown here as the area contained within the dashed lines) is a set of points whose nearest neighbor in X is u. The Delaunay graph is an undirected graph whose nodes are points in X and two nodes are connected (shown as solid lines) if their Voronoi regions have a non-empty intersection.
# 6.2.1 Voronoi Diagram
For the moment, suppose δ is the Euclidean distance and that we are in R2. Suppose further that we have a collection X of just two points u and v on the plane. Consider now the subset of R2 comprising of all the points to which u is the closest point from X . Similarity, we can identify the subset to which v is the closest point. These two subsets are, in fact, partitions of the plane and are separated by a lineâthe points on this line are equidistant to u and v. In other words, two points in R2 induce a partitioning of the plane where each partition is âownedâ by a point and describes the set of points that are closer to it than they are to the other point. | 2401.09350#207 | 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": 207,
"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. 6.2: Visualization of the Voronoi diagram (a) and its dual, the Delaunay graph (b) for an example collection X of points in R2. A Voronoi region associated with a point u (shown here as the area contained within the dashed lines) is a set of points whose nearest neighbor in X is u. The Delaunay graph is an undirected graph whose nodes are points in X and two nodes are connected (shown as solid lines) if their Voronoi regions have a non-empty intersection.\n# 6.2.1 Voronoi Diagram\nFor the moment, suppose δ is the Euclidean distance and that we are in R2. Suppose further that we have a collection X of just two points u and v on the plane. Consider now the subset of R2 comprising of all the points to which u is the closest point from X . Similarity, we can identify the subset to which v is the closest point. These two subsets are, in fact, partitions of the plane and are separated by a lineâthe points on this line are equidistant to u and v. In other words, two points in R2 induce a partitioning of the plane where each partition is âownedâ by a point and describes the set of points that are closer to it than they are to the other point.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 208 | We can trivially generalize that notion to more than two points and, in- deed, to higher dimensions. A collection ¥ of points in R¢ partitions the space into unique regions R = Uc Ru; where the region R,, is owned by point u ⬠¥ and represents the set of points to which wu is the closest point in X. Formally, Ry = {x | u = arg min, 0(a,v)}. Note that, each region is a convex polytope that is the intersection of half-spaces. The set of regions is known as the Voronoi diagram for the collection ¥ and is illustrated in Figure 6.2(a) for an example collection in R?.
77
78
6 Graph Algorithms
# 6.2.2 Delaunay Graph
The Delaunay graph for X is, in effect, a graph representation of its Voronoi diagram. The nodes of the graph are trivially the points in X , as before. We place an edge between two nodes u and v if their Voronoi regions have a non-empty intersection: Ru â© Rv ̸= â
. Clearly, by construction, this graph is undirected. An example of this graph is rendered in Figure 6.2(b). | 2401.09350#208 | 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": 208,
"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 can trivially generalize that notion to more than two points and, in- deed, to higher dimensions. A collection Â¥ of points in R¢ partitions the space into unique regions R = Uc Ru; where the region R,, is owned by point u ⬠¥ and represents the set of points to which wu is the closest point in X. Formally, Ry = {x | u = arg min, 0(a,v)}. Note that, each region is a convex polytope that is the intersection of half-spaces. The set of regions is known as the Voronoi diagram for the collection Â¥ and is illustrated in Figure 6.2(a) for an example collection in R?.\n77\n78\n6 Graph Algorithms\n# 6.2.2 Delaunay Graph\nThe Delaunay graph for X is, in effect, a graph representation of its Voronoi diagram. The nodes of the graph are trivially the points in X , as before. We place an edge between two nodes u and v if their Voronoi regions have a non-empty intersection: Ru â© Rv ̸= â
. Clearly, by construction, this graph is undirected. An example of this graph is rendered in Figure 6.2(b).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 209 | There is an important technical detail that is worth noting. The Delaunay graph for a collection X is unique if the points in X are in general posi- tion [Fortune, 1997]. A collection of points are said to be in general position if the following two conditions are satisfied. First, no n points from X â Rd, for 2 ⤠n ⤠d + 1, must lie on a (n â 2)-flat. Second, no n + 1 points must lie on any (nâ2)-dimensional hypersphere. In R2, as an example, for a collection of points to be in general position, no three points may be co-linear, and no four points co-circular.
We must add that, the detail above is generally satisfied in practice. Im- portantly, if the vectors in our collection are independent and identically distributed, then the collection is almost surely in general position. That is why we often take that technicality for granted. So from now on, we assume that the Delaunay graph of a collection of points is unique.
# 6.2.3 Top-1 Retrieval | 2401.09350#209 | 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": 209,
"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 important technical detail that is worth noting. The Delaunay graph for a collection X is unique if the points in X are in general posi- tion [Fortune, 1997]. A collection of points are said to be in general position if the following two conditions are satisfied. First, no n points from X â Rd, for 2 ⤠n ⤠d + 1, must lie on a (n â 2)-flat. Second, no n + 1 points must lie on any (nâ2)-dimensional hypersphere. In R2, as an example, for a collection of points to be in general position, no three points may be co-linear, and no four points co-circular.\nWe must add that, the detail above is generally satisfied in practice. Im- portantly, if the vectors in our collection are independent and identically distributed, then the collection is almost surely in general position. That is why we often take that technicality for granted. So from now on, we assume that the Delaunay graph of a collection of points is unique.\n# 6.2.3 Top-1 Retrieval",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 210 | # 6.2.3 Top-1 Retrieval
We can immediately recognize the importance of Voronoi regions: They ge- ometrically capture the set of queries for which a point from the collection is the solution to the top-1 retrieval problem. But what is the significance of the dual representation of this geometrical concept? How does the Delaunay graph help us solve the top-1 retrieval problem?
For one, the Delaunay graph is a compact representation of the Voronoi diagram. Instead of describing polytopes, we need only to record edges be- tween neighboring nodes. But, more crucially, as the following claim shows, we can traverse the Delaunay graph greedily, and reach the optimal top-1 solution from any node. In other words, the Delaunay graph has the desired property we described in Section 6.1.
Theorem 6.1 Let G = (V, E) be a graph that contains the Delaunay graph of m vectors X â Rd. The best-first-search algorithm over G gives the optimal solution to the top-1 retrieval problem for any arbitrary query q if δ(·, ·) is proper.
The proof of the result above relies on an important property of the De- launay graph, which we state first.
6.2 The Delaunay Graph | 2401.09350#210 | 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": 210,
"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": "# 6.2.3 Top-1 Retrieval\nWe can immediately recognize the importance of Voronoi regions: They ge- ometrically capture the set of queries for which a point from the collection is the solution to the top-1 retrieval problem. But what is the significance of the dual representation of this geometrical concept? How does the Delaunay graph help us solve the top-1 retrieval problem?\nFor one, the Delaunay graph is a compact representation of the Voronoi diagram. Instead of describing polytopes, we need only to record edges be- tween neighboring nodes. But, more crucially, as the following claim shows, we can traverse the Delaunay graph greedily, and reach the optimal top-1 solution from any node. In other words, the Delaunay graph has the desired property we described in Section 6.1.\nTheorem 6.1 Let G = (V, E) be a graph that contains the Delaunay graph of m vectors X â Rd. The best-first-search algorithm over G gives the optimal solution to the top-1 retrieval problem for any arbitrary query q if δ(·, ·) is proper.\nThe proof of the result above relies on an important property of the De- launay graph, which we state first.\n6.2 The Delaunay Graph",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 211 | Fig. 6.3: Illustration of the second case in the proof of Lemma 6.1.
Lemma 6.1 Let G = (V, E) be the Delaunay graph of a collection of points X â Rd, and let B be a ball centered at µ that contains two points u, v â X , with radius r = min(δ(µ, u), δ(µ, v)), for a continuous and proper distance function δ(·, ·). Then either (u, v) â E or there exists a third point in X that is contained in B.
Proof. Suppose there is no other point in X that is contained in B. We must show that, in that case, (u, v) â E.
There are two cases. The first and easy case is when u and v are on the surface of B. Clearly, u and v are equidistant from µ. Because there are no other points in B, we can conclude that µ lies in the intersection of Ru and Rv, the Voronoi regions associated with u and v. That implies (u, v) â E. | 2401.09350#211 | 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": 211,
"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. 6.3: Illustration of the second case in the proof of Lemma 6.1.\nLemma 6.1 Let G = (V, E) be the Delaunay graph of a collection of points X â Rd, and let B be a ball centered at µ that contains two points u, v â X , with radius r = min(δ(µ, u), δ(µ, v)), for a continuous and proper distance function δ(·, ·). Then either (u, v) â E or there exists a third point in X that is contained in B.\nProof. Suppose there is no other point in X that is contained in B. We must show that, in that case, (u, v) â E.\nThere are two cases. The first and easy case is when u and v are on the surface of B. Clearly, u and v are equidistant from µ. Because there are no other points in B, we can conclude that µ lies in the intersection of Ru and Rv, the Voronoi regions associated with u and v. That implies (u, v) â E.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 316 | Part III Compression
Chapter 9 Quantization
Abstract In a vector retrieval system, it is usually not enough to process queries as fast as possible. It is equally as important to reduce the size of the index by compressing vectors. Compression, however, must be done in such a way that either decompressing the vectors during retrieval incurs a negli- gible cost, or distances can be computed (approximately) in the compressed domain, rendering it unnecessary to decompress compressed vectors during retrieval. This chapter introduces a class of vector compression algorithms, known as quantization, that is inspired by clustering.
# 9.1 Vector Quantization | 2401.09350#316 | 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": 316,
"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": "Part III Compression\nChapter 9 Quantization\nAbstract In a vector retrieval system, it is usually not enough to process queries as fast as possible. It is equally as important to reduce the size of the index by compressing vectors. Compression, however, must be done in such a way that either decompressing the vectors during retrieval incurs a negli- gible cost, or distances can be computed (approximately) in the compressed domain, rendering it unnecessary to decompress compressed vectors during retrieval. This chapter introduces a class of vector compression algorithms, known as quantization, that is inspired by clustering.\n# 9.1 Vector Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 317 | # 9.1 Vector Quantization
Let us take a step back and present a different mental model of the clustering- based retrieval framework discussed in Chapter 7. At a high level, we band together points that are placed by ζ(·) into cluster i and represent that group by µi, for i â [C]. In the first stage of the search for query q, we take the following conceptual step: First, we compute δ(q, µi) for every i and construct a âtableâ that maps i to δ(q, µi). We next approximate δ(q, u) for every u â X using the resulting table: If u â ζ â1(i), then we look up an estimate of its distance to q from the i-th row of the table. We then identify the â closest distances, and perform a secondary search over the corresponding vectors. | 2401.09350#317 | 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": 317,
"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": "# 9.1 Vector Quantization\nLet us take a step back and present a different mental model of the clustering- based retrieval framework discussed in Chapter 7. At a high level, we band together points that are placed by ζ(·) into cluster i and represent that group by µi, for i â [C]. In the first stage of the search for query q, we take the following conceptual step: First, we compute δ(q, µi) for every i and construct a âtableâ that maps i to δ(q, µi). We next approximate δ(q, u) for every u â X using the resulting table: If u â ζ â1(i), then we look up an estimate of its distance to q from the i-th row of the table. We then identify the â closest distances, and perform a secondary search over the corresponding vectors.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 318 | This presentation of clustering for top-k retrieval highlights an important fact that does not come across as clearly in our original description of the algorithm: We have made an implicit assumption that δ(q, u) â δ(q, µi) for all u â ζ â1(i). That is why we presume that if a cluster minimizes δ(q, ·), then the points within it are also likely to minimize δ(q, ·). That is, in turn, why we deem it sufficient to search over the points within the top-â clusters. Put differently, within the first stage of search, we appear to be approx- imating every point u â ζ â1(i) with Ëu = µi. Because there are C discrete choices to consider for every data point, we can say that we quantize the
127
128
9 Quantization
vectors into [C]. Consequently, we can encode each vector using only log2 C bits, and an entire collection of vectors using m log2 C bits! All together, we can represent a collection X using O(Cd + m log2 C) space, and compute distances to a query by performing m look-ups into a table that itself takes O(Cd) time to construct. That quantity can be far smaller than O(md) given by the na¨ıve distance computation algorithm. | 2401.09350#318 | 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": 318,
"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": "This presentation of clustering for top-k retrieval highlights an important fact that does not come across as clearly in our original description of the algorithm: We have made an implicit assumption that δ(q, u) â δ(q, µi) for all u â ζ â1(i). That is why we presume that if a cluster minimizes δ(q, ·), then the points within it are also likely to minimize δ(q, ·). That is, in turn, why we deem it sufficient to search over the points within the top-â clusters. Put differently, within the first stage of search, we appear to be approx- imating every point u â ζ â1(i) with Ëu = µi. Because there are C discrete choices to consider for every data point, we can say that we quantize the\n127\n128\n9 Quantization\nvectors into [C]. Consequently, we can encode each vector using only log2 C bits, and an entire collection of vectors using m log2 C bits! All together, we can represent a collection X using O(Cd + m log2 C) space, and compute distances to a query by performing m look-ups into a table that itself takes O(Cd) time to construct. That quantity can be far smaller than O(md) given by the na¨ıve distance computation algorithm.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 319 | Clearly, the approximation error, â¥uâ Ëuâ¥, is a function of C. As we increase C, this approximation improves, so that â¥u â Ëu⥠â 0 and |δ(q, u) â δ(q, Ëu)| â 0. Indeed, C = m implies that Ëu = u for every u. But increasing C results in an increased space complexity and a less efficient distance computation. At C = m, for example, our table-building exercise does not help speed up distance computation for individual data pointsâbecause we must construct the table in O(md) time anyway. Finding the right C is therefore critical to space- and time-complexity, as well as the approximation or quantization error.
# 9.1.1 Codebooks and Codewords | 2401.09350#319 | 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": 319,
"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": "Clearly, the approximation error, â¥uâ Ëuâ¥, is a function of C. As we increase C, this approximation improves, so that â¥u â Ëu⥠â 0 and |δ(q, u) â δ(q, Ëu)| â 0. Indeed, C = m implies that Ëu = u for every u. But increasing C results in an increased space complexity and a less efficient distance computation. At C = m, for example, our table-building exercise does not help speed up distance computation for individual data pointsâbecause we must construct the table in O(md) time anyway. Finding the right C is therefore critical to space- and time-complexity, as well as the approximation or quantization error.\n# 9.1.1 Codebooks and Codewords",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 320 | # 9.1.1 Codebooks and Codewords
What we described above is known as vector quantization [Gray and Neuhoff, 1998] for vectors in the L2 space. We will therefore assume that δ(u, v) = 2 in the remainder of this section. The function ζ : Rd â [C] is called â¥u â vâ¥2 a quantizer, the individual centroids are referred to as codewords, and the set of C codewords make up a codebook. It is easy to see that the set ζ â1(i) is the intersection of X with the Voronoi region associated with codeword µi. The approximation quality of a given codebook is measured by the famil- iar mean squared error: E[â¥ÂµÎ¶(U ) â U â¥2 2)], with U denoting a random vector. Interestingly, that is exactly the objective that is minimized by Lloydâs al- gorithm for KMeans clustering. As such, an optimal codebook is one that satisfies Lloydâs optimality conditions: each data point must be quantized to its nearest codeword, and each Voronoi region must be represented by its mean. That is why KMeans is our default choice for ζ.
# 9.2 Product Quantization | 2401.09350#320 | 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": 320,
"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": "# 9.1.1 Codebooks and Codewords\nWhat we described above is known as vector quantization [Gray and Neuhoff, 1998] for vectors in the L2 space. We will therefore assume that δ(u, v) = 2 in the remainder of this section. The function ζ : Rd â [C] is called â¥u â vâ¥2 a quantizer, the individual centroids are referred to as codewords, and the set of C codewords make up a codebook. It is easy to see that the set ζ â1(i) is the intersection of X with the Voronoi region associated with codeword µi. The approximation quality of a given codebook is measured by the famil- iar mean squared error: E[â¥ÂµÎ¶(U ) â U â¥2 2)], with U denoting a random vector. Interestingly, that is exactly the objective that is minimized by Lloydâs al- gorithm for KMeans clustering. As such, an optimal codebook is one that satisfies Lloydâs optimality conditions: each data point must be quantized to its nearest codeword, and each Voronoi region must be represented by its mean. That is why KMeans is our default choice for ζ.\n# 9.2 Product Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 321 | # 9.2 Product Quantization
As we noted earlier, the quantization error is a function of the number of clusters, C: A larger value of C drives down the approximation error, making the quantization and the subsequent top-k retrieval solution more accurate and effective. However, realistically, C cannot become too large, because then the framework would collapse to exhaustive search, degrading its efficiency. How may we reconcile the two seemingly opposing forces?
9.2 Product Quantization
J´egou et al. [2011] gave an answer to that question in the form of Product Quantization (PQ). The idea is easy to describe at a high level: Whereas in vector quantization we quantize the entire vector into one of C clusters, in PQ we break up a vector into orthogonal subspaces and perform vector quantization on individual chunks separately. The quantized vector is then a concatenation of the quantized subspaces. | 2401.09350#321 | 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": 321,
"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": "# 9.2 Product Quantization\nAs we noted earlier, the quantization error is a function of the number of clusters, C: A larger value of C drives down the approximation error, making the quantization and the subsequent top-k retrieval solution more accurate and effective. However, realistically, C cannot become too large, because then the framework would collapse to exhaustive search, degrading its efficiency. How may we reconcile the two seemingly opposing forces?\n9.2 Product Quantization\nJ´egou et al. [2011] gave an answer to that question in the form of Product Quantization (PQ). The idea is easy to describe at a high level: Whereas in vector quantization we quantize the entire vector into one of C clusters, in PQ we break up a vector into orthogonal subspaces and perform vector quantization on individual chunks separately. The quantized vector is then a concatenation of the quantized subspaces.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 322 | Formally, suppose that the number of dimensions d is divisible by dâ¦, and let L = d/dâ¦. Define a selector matrix Si â {0, 1}dâ¦Ãd, 1 ⤠i ⤠L as a matrix with L blocks in {0, 1}dâ¦Ãd⦠, where all blocks are 0 but the i-th block is the identity. The following is an example for d = 6, d⦠= 2, and i = 2:
g, â 001000 21000100
For a given vector u ⬠R¢, S;u gives the i-th d.-dimensional subspace, so that we can write: u = @, Su. Suppose further that we have n quantizers G through ¢;, where ¢; : R¢° â [C] maps the subspace selected by S$; to one of C clusters. Each ¢; gives us C centroids ju;,; for 7 ⬠[C]. | 2401.09350#322 | 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": 322,
"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": "Formally, suppose that the number of dimensions d is divisible by dâ¦, and let L = d/dâ¦. Define a selector matrix Si â {0, 1}dâ¦Ãd, 1 ⤠i ⤠L as a matrix with L blocks in {0, 1}dâ¦Ãd⦠, where all blocks are 0 but the i-th block is the identity. The following is an example for d = 6, d⦠= 2, and i = 2:\ng, â 001000 21000100\nFor a given vector u ⬠R¢, S;u gives the i-th d.-dimensional subspace, so that we can write: u = @, Su. Suppose further that we have n quantizers G through ¢;, where ¢; : R¢° â [C] maps the subspace selected by S$; to one of C clusters. Each ¢; gives us C centroids ju;,; for 7 ⬠[C].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 323 | Using the notation above, we can express the PQ code for a vector u as L cluster identifiers, ζi(Siu), for i â [L]. We can therefore quantize a d- dimensional vector using L log2 C bits. Observe that, when L = 1 (or equiva- lently, d⦠= d), PQ reduces to vector quantization. When L = d, on the other hand, PQ performs scalar quantization per dimension.
Given this scheme, our approximation of u is &@ = Dj
i µi,ζi(u). It is easy to see that the quantization error E[â¥U â ËU â¥2 2], with U denoting a random vector drawn from X and ËU its reconstruction, is the sum of the quantization error of individual subspaces:
L EIIV â 813) = =F [lu- Brrccoold] i=l nex L 1 / . = > [dolsiw - H¢5(0) 3] uEX i=1
As a result, learning the L codebooks can be formulated as L independent sub-problems. The i-th codebook can therefore be learnt by the application of KMeans on SiX = {Siu | u â X }.
# 9.2.1 Distance Computation with PQ | 2401.09350#323 | 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": 323,
"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 notation above, we can express the PQ code for a vector u as L cluster identifiers, ζi(Siu), for i â [L]. We can therefore quantize a d- dimensional vector using L log2 C bits. Observe that, when L = 1 (or equiva- lently, d⦠= d), PQ reduces to vector quantization. When L = d, on the other hand, PQ performs scalar quantization per dimension.\nGiven this scheme, our approximation of u is &@ = Dj\ni µi,ζi(u). It is easy to see that the quantization error E[â¥U â ËU â¥2 2], with U denoting a random vector drawn from X and ËU its reconstruction, is the sum of the quantization error of individual subspaces:\nL EIIV â 813) = =F [lu- Brrccoold] i=l nex L 1 / . = > [dolsiw - H¢5(0) 3] uEX i=1\nAs a result, learning the L codebooks can be formulated as L independent sub-problems. The i-th codebook can therefore be learnt by the application of KMeans on SiX = {Siu | u â X }.\n# 9.2.1 Distance Computation with PQ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 324 | # 9.2.1 Distance Computation with PQ
In vector quantization, computing the distance of a vector u to a query q was fairly trivial. All we had to do was to precompute a table that maps i â [C] to â¥q â µiâ¥2, then look up the entry that corresponds to ζ(u). The fact that
129
130
9 Quantization
we were able to precompute C distances once per query, then simply look up the right entry from the table for a vector u helped us save a great deal of computation. Can we devise a similar algorithm given a PQ code?
The answer is yes. Indeed, that is why PQ has proven to be an efficient algorithm for distance computation. As in vector quantization, it first com- putes L distance tables, but the i-th table maps j â [C] to â¥Siq â µi,jâ¥2 2 (note the squared L2 distance). Using these tables, we can estimate the distance between q and any vector u as follows: | 2401.09350#324 | 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": 324,
"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": "# 9.2.1 Distance Computation with PQ\nIn vector quantization, computing the distance of a vector u to a query q was fairly trivial. All we had to do was to precompute a table that maps i â [C] to â¥q â µiâ¥2, then look up the entry that corresponds to ζ(u). The fact that\n129\n130\n9 Quantization\nwe were able to precompute C distances once per query, then simply look up the right entry from the table for a vector u helped us save a great deal of computation. Can we devise a similar algorithm given a PQ code?\nThe answer is yes. Indeed, that is why PQ has proven to be an efficient algorithm for distance computation. As in vector quantization, it first com- putes L distance tables, but the i-th table maps j â [C] to â¥Siq â µi,jâ¥2 2 (note the squared L2 distance). Using these tables, we can estimate the distance between q and any vector u as follows:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 325 | lla â ula = lla â al L = Na - Deicwyll i=1 (Sia ~ Hic) IB ae i=l L = Dillsia = Hie(u)lldObserve that, we have already computed the summands and recorded them in the distance tables. As a result, approximating the distance between u and q amounts to L table look-ups. The overall amount of computation needed to approximate distances between q and m vectors in X is then O(LCd⦠+ mL).
We must remark on the newly-introduced parameter dâ¦. Even though in the context of vector quantization, the impact of C on the quantization error is not theoretically known, there is nonetheless a clear interpretation: A larger C leads to better quantization. In PQ, the impact of d⦠or, equivalently, L on the quantization error is not as clear. As noted earlier, we can say something about d⦠at the extremes, but what we should expect from a value somewhere between 1 and d is largely an empirical question [Sun et al., 2023].
# 9.2.2 Optimized Product Quantization | 2401.09350#325 | 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": 325,
"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": "lla â ula = lla â al L = Na - Deicwyll i=1 (Sia ~ Hic) IB ae i=l L = Dillsia = Hie(u)lldObserve that, we have already computed the summands and recorded them in the distance tables. As a result, approximating the distance between u and q amounts to L table look-ups. The overall amount of computation needed to approximate distances between q and m vectors in X is then O(LCd⦠+ mL).\nWe must remark on the newly-introduced parameter dâ¦. Even though in the context of vector quantization, the impact of C on the quantization error is not theoretically known, there is nonetheless a clear interpretation: A larger C leads to better quantization. In PQ, the impact of d⦠or, equivalently, L on the quantization error is not as clear. As noted earlier, we can say something about d⦠at the extremes, but what we should expect from a value somewhere between 1 and d is largely an empirical question [Sun et al., 2023].\n# 9.2.2 Optimized Product Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 326 | # 9.2.2 Optimized Product Quantization
In PQ, we allocate an equal number of bits (log2 C) to each of the n orthog- onal subspaces. This makes sense if our vectors have similar energy in every subspace. But when the dimensions in one subspace are highly correlated, and in another uncorrolated, our equal-bits-per-subspace allocation policy proves wasteful in the former and perhaps inadequate in the latter. How can we ensure a more balanced energy across subspaces?
J´egou et al. [2011] argue that applying a random rotation R â RdÃd (RRT = I) to the data points prior to quantization is one way to reduce the correlation between dimensions. The matrix R together with Siâs, as
9.2 Product Quantization
defined above, determines how we decompose the vector space into its sub- spaces. By applying a rotation first, we no longer chunk up an input vector into sub-vectors that comprise of consecutive dimensions. | 2401.09350#326 | 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": 326,
"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": "# 9.2.2 Optimized Product Quantization\nIn PQ, we allocate an equal number of bits (log2 C) to each of the n orthog- onal subspaces. This makes sense if our vectors have similar energy in every subspace. But when the dimensions in one subspace are highly correlated, and in another uncorrolated, our equal-bits-per-subspace allocation policy proves wasteful in the former and perhaps inadequate in the latter. How can we ensure a more balanced energy across subspaces?\nJ´egou et al. [2011] argue that applying a random rotation R â RdÃd (RRT = I) to the data points prior to quantization is one way to reduce the correlation between dimensions. The matrix R together with Siâs, as\n9.2 Product Quantization\ndefined above, determines how we decompose the vector space into its sub- spaces. By applying a rotation first, we no longer chunk up an input vector into sub-vectors that comprise of consecutive dimensions.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 327 | Later, Ge et al. [2014] and Norouzi and Fleet [2013] extended this idea and suggested that the matrix R can be learnt jointly with the codebooks. This can be done through an iterative algorithm that switches between two steps in each iteration. In the first step, we freeze R and learn a PQ codebook as before. In the second step, we freeze the codebook and update the matrix R by solving the following optimization problem:
min R â¥Ru â Ëuâ¥2 2, uâX s.t. RRT = I,
where Ëu is the approximation of u according to the frozen PQ codebook. Because u and Ëu are fixed in the above optimization problem, we can rewrite the objective as follows:
min R s.t. RRT = I,
where U is a d-by-m matrix where each column is a vector in X , ËU is a matrix where each column is an approximation of the corresponding column in U , and â¥Â·â¥F is the Frobenius norm. This problem has a closed-form solution as shown by Ge et al. [2014].
# 9.2.3 Extensions | 2401.09350#327 | 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": 327,
"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": "Later, Ge et al. [2014] and Norouzi and Fleet [2013] extended this idea and suggested that the matrix R can be learnt jointly with the codebooks. This can be done through an iterative algorithm that switches between two steps in each iteration. In the first step, we freeze R and learn a PQ codebook as before. In the second step, we freeze the codebook and update the matrix R by solving the following optimization problem:\nmin R â¥Ru â Ëuâ¥2 2, uâX s.t. RRT = I,\nwhere Ëu is the approximation of u according to the frozen PQ codebook. Because u and Ëu are fixed in the above optimization problem, we can rewrite the objective as follows:\nmin R s.t. RRT = I,\nwhere U is a d-by-m matrix where each column is a vector in X , ËU is a matrix where each column is an approximation of the corresponding column in U , and â¥Â·â¥F is the Frobenius norm. This problem has a closed-form solution as shown by Ge et al. [2014].\n# 9.2.3 Extensions",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 328 | # 9.2.3 Extensions
Since the study by J´egou et al. [2011], many variations of the idea have emerged in the literature. In the original publication, for example, J´egou et al. [2011] used PQ codes in conjunction with the clustering-based retrieval framework presented earlier in this chapter. In other words, a collection X is first clustered into C clusters (âcoarse-quantizationâ), and each cluster is subsequently represented using its own PQ codebook. In this way, when the routing function identifies a cluster to search, we can compute distances for data points within that cluster using their PQ codes. Later, Babenko and Lempitsky [2012] extended this two-level quantization further by introducing the âinverted multi-indexâ structure.
When combining PQ with clustering or coarse-quantization, instead of producing PQ codebooks for raw vectors within each cluster, one could learn codebooks for the residual vectors instead. That means, if the centroid of the i-th cluster is µi, then we may quantize (u â µi) for each vector u â ζ â1(i).
131
132
9 Quantization | 2401.09350#328 | 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": 328,
"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": "# 9.2.3 Extensions\nSince the study by J´egou et al. [2011], many variations of the idea have emerged in the literature. In the original publication, for example, J´egou et al. [2011] used PQ codes in conjunction with the clustering-based retrieval framework presented earlier in this chapter. In other words, a collection X is first clustered into C clusters (âcoarse-quantizationâ), and each cluster is subsequently represented using its own PQ codebook. In this way, when the routing function identifies a cluster to search, we can compute distances for data points within that cluster using their PQ codes. Later, Babenko and Lempitsky [2012] extended this two-level quantization further by introducing the âinverted multi-indexâ structure.\nWhen combining PQ with clustering or coarse-quantization, instead of producing PQ codebooks for raw vectors within each cluster, one could learn codebooks for the residual vectors instead. That means, if the centroid of the i-th cluster is µi, then we may quantize (u â µi) for each vector u â ζ â1(i).\n131\n132\n9 Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 329 | 131
132
9 Quantization
This was the idea first introduced by J´egou et al. [2011], then developed further in subsequent works [Kalantidis and Avrithis, 2014, Wu et al., 2017]. The PQ literature does not end there. In fact, so popular, effective, and efficient is PQ that it pops up in many different contexts and a variety of applications. Research into improving its accuracy and speed is still ongoing. For example, there have been many works that speed up the distance com- putation with PQ codebooks by leveraging hardware capabilities [Johnson et al., 2021, Andre et al., 2021, Andr´e et al., 2015]. Others that extend the algorithm to streaming (online) collections [Xu et al., 2018], and yet other studies that investigate other PQ codebook-learning protocols [Liu et al., 2020, Yu et al., 2018, Chen et al., 2020, Jang and Cho, 2021, Klein and Wolf, 2019, Lu et al., 2023]. This list is certainly not exhaustive and is still growing.
# 9.3 Additive Quantization | 2401.09350#329 | 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": 329,
"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": "131\n132\n9 Quantization\nThis was the idea first introduced by J´egou et al. [2011], then developed further in subsequent works [Kalantidis and Avrithis, 2014, Wu et al., 2017]. The PQ literature does not end there. In fact, so popular, effective, and efficient is PQ that it pops up in many different contexts and a variety of applications. Research into improving its accuracy and speed is still ongoing. For example, there have been many works that speed up the distance com- putation with PQ codebooks by leveraging hardware capabilities [Johnson et al., 2021, Andre et al., 2021, Andr´e et al., 2015]. Others that extend the algorithm to streaming (online) collections [Xu et al., 2018], and yet other studies that investigate other PQ codebook-learning protocols [Liu et al., 2020, Yu et al., 2018, Chen et al., 2020, Jang and Cho, 2021, Klein and Wolf, 2019, Lu et al., 2023]. This list is certainly not exhaustive and is still growing.\n# 9.3 Additive Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 330 | # 9.3 Additive Quantization
PQ remains the dominant quantization method for top-k retrieval due to its overall simplicity and the efficiency of its codebook learning protocol. There are, however, numerous generalizations of the framework [Babenko and Lempitsky, 2014, Chen et al., 2010, Niu et al., 2023, Liu et al., 2015, Ozan et al., 2016, Krishnan and Liberty, 2021]. Typically, these generalized forms improve the approximation error but require more involved codebook learning algorithms and vector encoding protocols. In this section, we review one key algorithm, known as Additive Quantization (AQ) [Babenko and Lempitsky, 2014], that is the backbone of all other methods.
Like PQ, AQ learns L codebooks where each codebook consists of C code- words. Unlike PQ, however, each codeword is a vector in R¢ârather than R*. Furthermore, a vector u is approximated as the sum, instead of the concatenation, of L codewords, one from each codebook: & = ean Hi,ci(u)s where ¢; : R¢ â [C] is the quantizer associated with the i-th codebook. | 2401.09350#330 | 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": 330,
"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": "# 9.3 Additive Quantization\nPQ remains the dominant quantization method for top-k retrieval due to its overall simplicity and the efficiency of its codebook learning protocol. There are, however, numerous generalizations of the framework [Babenko and Lempitsky, 2014, Chen et al., 2010, Niu et al., 2023, Liu et al., 2015, Ozan et al., 2016, Krishnan and Liberty, 2021]. Typically, these generalized forms improve the approximation error but require more involved codebook learning algorithms and vector encoding protocols. In this section, we review one key algorithm, known as Additive Quantization (AQ) [Babenko and Lempitsky, 2014], that is the backbone of all other methods.\nLike PQ, AQ learns L codebooks where each codebook consists of C code- words. Unlike PQ, however, each codeword is a vector in R¢ârather than R*. Furthermore, a vector u is approximated as the sum, instead of the concatenation, of L codewords, one from each codebook: & = ean Hi,ci(u)s where ¢; : R¢ â [C] is the quantizer associated with the i-th codebook.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 331 | Let us compare AQ with PQ at a high level and understand how AQ is different. We can still encode a data point using L log2 C bits, as in PQ. How- ever, the codebooks for AQ are L-times larger than their PQ counterparts, simply because each codeword has d dimensions instead of dâ¦. On the other hand, AQ does not decompose the space into orthogonal subspaces and, as such, makes no assumptions about the independence between subspaces.
AQ is therefore a strictly more general quantization method than PQ. In fact, the class of additive quantizers contains the class of product quantizers: By restricting the i-th codebook in AQ to the set of codewords that are 0 everywhere outside of the i-th âchunk,â we recover PQ. Empirical compar- isons [Babenko and Lempitsky, 2014, Matsui et al., 2018] confirm that such a generalization is more effective in practice.
For this formulation to be complete, we have to specify how the codebooks are learnt, how we encode an arbitrary vector, and how we perform distance
9.3 Additive Quantization
computation. We will cover these topics in reverse order in the following sections.
# 9.3.1 Distance Computation with AQ | 2401.09350#331 | 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": 331,
"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 compare AQ with PQ at a high level and understand how AQ is different. We can still encode a data point using L log2 C bits, as in PQ. How- ever, the codebooks for AQ are L-times larger than their PQ counterparts, simply because each codeword has d dimensions instead of dâ¦. On the other hand, AQ does not decompose the space into orthogonal subspaces and, as such, makes no assumptions about the independence between subspaces.\nAQ is therefore a strictly more general quantization method than PQ. In fact, the class of additive quantizers contains the class of product quantizers: By restricting the i-th codebook in AQ to the set of codewords that are 0 everywhere outside of the i-th âchunk,â we recover PQ. Empirical compar- isons [Babenko and Lempitsky, 2014, Matsui et al., 2018] confirm that such a generalization is more effective in practice.\nFor this formulation to be complete, we have to specify how the codebooks are learnt, how we encode an arbitrary vector, and how we perform distance\n9.3 Additive Quantization\ncomputation. We will cover these topics in reverse order in the following sections.\n# 9.3.1 Distance Computation with AQ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 332 | 9.3 Additive Quantization
computation. We will cover these topics in reverse order in the following sections.
# 9.3.1 Distance Computation with AQ
Suppose for the moment that we have learnt AQ codebooks for a collection X and that we are able to encode an arbitrary vector into an AQ code (i.e., a vector of L codeword identifiers). In this section, we examine how we may compute the distance between a query point q and a data point u using its approximation Ëu.
Observe the following fact:
â¥q â uâ¥2 2 = â¥qâ¥2 2 â 2â¨q, uâ© + â¥uâ¥2 2.
The first term is a constant that can be computed once per query and, at any rate, is inconsequential to the top-k retrieval problem. The last term, â¥uâ¥2 2 can be stored for every vector and looked up during distance computation, as suggested by Babenko and Lempitsky [2014]. That means, the encoding of a vector u â X comprises of two components: Ëu and its (possibly scalar- quantized) squared norm. This brings the total space required to encode m vectors to O(LCd + m(1 + L log2 C)). | 2401.09350#332 | 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": 332,
"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": "9.3 Additive Quantization\ncomputation. We will cover these topics in reverse order in the following sections.\n# 9.3.1 Distance Computation with AQ\nSuppose for the moment that we have learnt AQ codebooks for a collection X and that we are able to encode an arbitrary vector into an AQ code (i.e., a vector of L codeword identifiers). In this section, we examine how we may compute the distance between a query point q and a data point u using its approximation Ëu.\nObserve the following fact:\nâ¥q â uâ¥2 2 = â¥qâ¥2 2 â 2â¨q, uâ© + â¥uâ¥2 2.\nThe first term is a constant that can be computed once per query and, at any rate, is inconsequential to the top-k retrieval problem. The last term, â¥uâ¥2 2 can be stored for every vector and looked up during distance computation, as suggested by Babenko and Lempitsky [2014]. That means, the encoding of a vector u â X comprises of two components: Ëu and its (possibly scalar- quantized) squared norm. This brings the total space required to encode m vectors to O(LCd + m(1 + L log2 C)).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 333 | The middle term can be approximated by â¨q, Ëuâ© and can be expressed as follows:
â¨q, uâ© â â¨q, Ëuâ© = â¨q, µi,ζi(u)â©.
As in PQ, the summands can be computed once for all codewords, and stored in a table. When approximating the inner product, we can do as before and look up the appropriate entries from these precomputed tables. The time complexity of this operation is therefore O(LCd + mL) for m data points, which is similar to PQ.
# 9.3.2 AQ Encoding and Codebook Learning
While distance computation with AQ codes is fairly similar to the process involving PQ codes, the encoding of a data point is substantially different and relatively complex in AQ. That is because we can no longer simply assign a vector to its nearest codeword. Instead, we must find an arrangement of L codewords that together minimize the approximation error â¥u â Ëuâ¥2.
Let us expand the expression for the approximation error as follows:
133
134
9 Quantization | 2401.09350#333 | 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": 333,
"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 middle term can be approximated by â¨q, Ëuâ© and can be expressed as follows:\nâ¨q, uâ© â â¨q, Ëuâ© = â¨q, µi,ζi(u)â©.\nAs in PQ, the summands can be computed once for all codewords, and stored in a table. When approximating the inner product, we can do as before and look up the appropriate entries from these precomputed tables. The time complexity of this operation is therefore O(LCd + mL) for m data points, which is similar to PQ.\n# 9.3.2 AQ Encoding and Codebook Learning\nWhile distance computation with AQ codes is fairly similar to the process involving PQ codes, the encoding of a data point is substantially different and relatively complex in AQ. That is because we can no longer simply assign a vector to its nearest codeword. Instead, we must find an arrangement of L codewords that together minimize the approximation error â¥u â Ëuâ¥2.\nLet us expand the expression for the approximation error as follows:\n133\n134\n9 Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 334 | Let us expand the expression for the approximation error as follows:
133
134
9 Quantization
L ju â GI)3 = lu â SP pircacuylld i=l Nw L L = llulls = 20, 7 piceeuy) + IDE Hncccny | i=l i=1 L = [ul + (D3 â2¢u, wiccscuy) + llccsc 1 i= SE Hiccups Hi.g(0))- 1<i<j<L N 3)+
Notice that the first term is irrelevant to the objective function, so we may ignore it. We must therefore find ζiâs that minimize the remaining terms.
Babenko and Lempitsky [2014] use a generalized Beam search to solve this optimization problem. The algorithm begins by selecting L closest code- words from Uy fei ...fi,c} to u. For a chosen codeword juy,;, we com- pute the residual u â pz,; and find the L closest codewords to it from Uien (Hi ..-fi,c}. After performing this search for all chosen codewords from the first round, we end up with a maximum of L? unique pairs of code- words. Note that, each pair has codewords from two different codebooks. | 2401.09350#334 | 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": 334,
"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 expand the expression for the approximation error as follows:\n133\n134\n9 Quantization\nL ju â GI)3 = lu â SP pircacuylld i=l Nw L L = llulls = 20, 7 piceeuy) + IDE Hncccny | i=l i=1 L = [ul + (D3 â2¢u, wiccscuy) + llccsc 1 i= SE Hiccups Hi.g(0))- 1<i<j<L N 3)+\nNotice that the first term is irrelevant to the objective function, so we may ignore it. We must therefore find ζiâs that minimize the remaining terms.\nBabenko and Lempitsky [2014] use a generalized Beam search to solve this optimization problem. The algorithm begins by selecting L closest code- words from Uy fei ...fi,c} to u. For a chosen codeword juy,;, we com- pute the residual u â pz,; and find the L closest codewords to it from Uien (Hi ..-fi,c}. After performing this search for all chosen codewords from the first round, we end up with a maximum of L? unique pairs of code- words. Note that, each pair has codewords from two different codebooks.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 335 | Of the L2 pairs, the algorithm picks the top L that minimize the approx- imation error. It then repeats this process for a total of L rounds, where in each round we compute the residuals given L tuples of codewords, and for each tuple, find L codewords from the remaining codebooks, and ultimately identify the top L tuples from the L2 tuples. At the end of the L-th round, the tuple with the minimal approximation error is the encoding for u.
Now that we have addressed the vector encoding part, it remains to de- scribe the codebook learning procedure. Unsurprisingly, learning a codebook is not so dissimilar to the PQ codebook learning algorithm. It is an iterative procedure alternating between two steps to optimize the following objective:
L min S> ju â SO pi.cicwy l- i=l
One step of every iteration freezes the codewords and performs assignments ζiâs, which is the encoding problem we have already discussed above. The second step freezes the assignments and updates the codewords, which itself is a least-squares problem that can be solved relatively efficiently, considering that it decomposes over each dimension.
9.4 Quantization for Inner Product
# 9.4 Quantization for Inner Product | 2401.09350#335 | 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": 335,
"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 the L2 pairs, the algorithm picks the top L that minimize the approx- imation error. It then repeats this process for a total of L rounds, where in each round we compute the residuals given L tuples of codewords, and for each tuple, find L codewords from the remaining codebooks, and ultimately identify the top L tuples from the L2 tuples. At the end of the L-th round, the tuple with the minimal approximation error is the encoding for u.\nNow that we have addressed the vector encoding part, it remains to de- scribe the codebook learning procedure. Unsurprisingly, learning a codebook is not so dissimilar to the PQ codebook learning algorithm. It is an iterative procedure alternating between two steps to optimize the following objective:\nL min S> ju â SO pi.cicwy l- i=l\nOne step of every iteration freezes the codewords and performs assignments ζiâs, which is the encoding problem we have already discussed above. The second step freezes the assignments and updates the codewords, which itself is a least-squares problem that can be solved relatively efficiently, considering that it decomposes over each dimension.\n9.4 Quantization for Inner Product\n# 9.4 Quantization for Inner Product",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 336 | 9.4 Quantization for Inner Product
# 9.4 Quantization for Inner Product
The vector quantization literature has largely been focused on the Euclidean distance and the approximate nearest neighbor search problem. Those ideas typically port over to the maximum cosine similarity search with little effort, but not to MIPS under general conditions. To understand why, suppose we wish to find a quantizer such that the inner product approximation error is minimized for a query distribution: E p> (a, u) â (4, a)â | = DD E (a, ur i?)
E p> (a, u) â (4, a)â | = DD E (a, ur i?) UucxX ue =SCE [(w â a)" qq" (uâ i) qd uEx = Di (u- TE [aa"] (uâ 4%), (9.1) uEx | 2401.09350#336 | 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": 336,
"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": "9.4 Quantization for Inner Product\n# 9.4 Quantization for Inner Product\nThe vector quantization literature has largely been focused on the Euclidean distance and the approximate nearest neighbor search problem. Those ideas typically port over to the maximum cosine similarity search with little effort, but not to MIPS under general conditions. To understand why, suppose we wish to find a quantizer such that the inner product approximation error is minimized for a query distribution: E p> (a, u) â (4, a)â | = DD E (a, ur i?)\nE p> (a, u) â (4, a)â | = DD E (a, ur i?) UucxX ue =SCE [(w â a)\" qq\" (uâ i) qd uEx = Di (u- TE [aa\"] (uâ 4%), (9.1) uEx",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 337 | where Ëu is an approximation of u. If we assumed that q is isotropic, so that its covariance matrix is the identity matrix scaled by some constant, then the objective above reduces to the reconstruction error. In that particular case, it makes sense for the quantization objective to be based on the reconstruction error, making the quantization methods we have studied thus far appropriate for MIPS too. But in the more general case, where the distribution of q is anisotropic, there is a gap between the true objective and the reconstruction error.
Guo et al. [2016] showed that, if we are able to obtain a small sample of queries to estimate E[qqT ], then we can modify the assignment step in Lloydâs iterative algorithm for KMeans in order to minimize the objective in Equa- tion (9.1). That is, instead of assigning points to clusters by their Euclidean distance to the (frozen) centroids, we must instead use Mahalanobis distance characterized by E[qqT ]. The resulting quantizer is arguably more suitable for inner product than the plain reconstruction error.
# 9.4.1 Score-aware Quantization | 2401.09350#337 | 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": 337,
"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": "where Ëu is an approximation of u. If we assumed that q is isotropic, so that its covariance matrix is the identity matrix scaled by some constant, then the objective above reduces to the reconstruction error. In that particular case, it makes sense for the quantization objective to be based on the reconstruction error, making the quantization methods we have studied thus far appropriate for MIPS too. But in the more general case, where the distribution of q is anisotropic, there is a gap between the true objective and the reconstruction error.\nGuo et al. [2016] showed that, if we are able to obtain a small sample of queries to estimate E[qqT ], then we can modify the assignment step in Lloydâs iterative algorithm for KMeans in order to minimize the objective in Equa- tion (9.1). That is, instead of assigning points to clusters by their Euclidean distance to the (frozen) centroids, we must instead use Mahalanobis distance characterized by E[qqT ]. The resulting quantizer is arguably more suitable for inner product than the plain reconstruction error.\n# 9.4.1 Score-aware Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 338 | # 9.4.1 Score-aware Quantization
Later, Guo et al. [2020] argued that the objective in Equation (9.1) does not adequately capture the nuances of MIPS. Their argument rests on an observation and an intuition. The observation is that, in Equation (9.1), every single data point contributes equally to the optimization objective. Intuitively, however, data points are not equally likely to be the solution to MIPS. The error from data points that are more likely to be the maximizers of inner product with queries should therefore be weighted more heavily than others.
135
136
9 Quantization | 2401.09350#338 | 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": 338,
"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": "# 9.4.1 Score-aware Quantization\nLater, Guo et al. [2020] argued that the objective in Equation (9.1) does not adequately capture the nuances of MIPS. Their argument rests on an observation and an intuition. The observation is that, in Equation (9.1), every single data point contributes equally to the optimization objective. Intuitively, however, data points are not equally likely to be the solution to MIPS. The error from data points that are more likely to be the maximizers of inner product with queries should therefore be weighted more heavily than others.\n135\n136\n9 Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 339 | Fig. 9.1: Decomposition of the residual error r(u, Ëu) = u â Ëu for u â R2 to one component that is parallel to the data point, râ¥(u, Ëu), and another that is orthogonal to it, râ¥(u, Ëu).
On the basis of that argument, Guo et al. [2020] introduce the following objective for inner product quantization:
DE [oU(a.w)) (uw) . (9.2)
In the above, Ï : R â R+ is an arbitrary weight function that determines the importance of each data point to the optimization objective. Ideally, then, Ï should be monotonically non-decreasing in its argument. One such weight function is Ï(s) = 1sâ¥Î¸ for some threshold θ, implying that only data points whose expected inner product is at least θ contribute to the objective, while the rest are simply ignored. That is the weight function that Guo et al. [2020] choose in their work. | 2401.09350#339 | 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": 339,
"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. 9.1: Decomposition of the residual error r(u, Ëu) = u â Ëu for u â R2 to one component that is parallel to the data point, râ¥(u, Ëu), and another that is orthogonal to it, râ¥(u, Ëu).\nOn the basis of that argument, Guo et al. [2020] introduce the following objective for inner product quantization:\nDE [oU(a.w)) (uw) . (9.2)\nIn the above, Ï : R â R+ is an arbitrary weight function that determines the importance of each data point to the optimization objective. Ideally, then, Ï should be monotonically non-decreasing in its argument. One such weight function is Ï(s) = 1sâ¥Î¸ for some threshold θ, implying that only data points whose expected inner product is at least θ contribute to the objective, while the rest are simply ignored. That is the weight function that Guo et al. [2020] choose in their work.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 340 | Something interesting emerges from Equation (9.2) with the choice of Ï(s) = 1sâ¥Î¸: It is more important for Ëu to preserve the norm of u than it is to preserve its angle. We will show why that is shortly, but consider for the moment the reason this behavior is important for MIPS. Suppose there is a data point whose norm is much larger than the rest of the data points. Intuitively, such a data point has a good chance of maximizing inner prod- uct with a query even if its angle with the query is relatively large. In other words, being a candidate solution to MIPS is less sensitive to angles and more sensitive to norms. Of course, as norms become more and more concentrated, angles take on a bigger role in determining the solution to MIPS. So, intu- itively, an objective that penalizes the distortion of norms more than angles is more suitable for MIPS.
# 9.4.1.1 Parallel and Orthogonal Residuals
Let us present this phenomenon more formally and show why the statement above is true. Define the residual error as r(u, Ëu) = u â Ëu. The residual error
9.4 Quantization for Inner Product | 2401.09350#340 | 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": 340,
"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": "Something interesting emerges from Equation (9.2) with the choice of Ï(s) = 1sâ¥Î¸: It is more important for Ëu to preserve the norm of u than it is to preserve its angle. We will show why that is shortly, but consider for the moment the reason this behavior is important for MIPS. Suppose there is a data point whose norm is much larger than the rest of the data points. Intuitively, such a data point has a good chance of maximizing inner prod- uct with a query even if its angle with the query is relatively large. In other words, being a candidate solution to MIPS is less sensitive to angles and more sensitive to norms. Of course, as norms become more and more concentrated, angles take on a bigger role in determining the solution to MIPS. So, intu- itively, an objective that penalizes the distortion of norms more than angles is more suitable for MIPS.\n# 9.4.1.1 Parallel and Orthogonal Residuals\nLet us present this phenomenon more formally and show why the statement above is true. Define the residual error as r(u, Ëu) = u â Ëu. The residual error\n9.4 Quantization for Inner Product",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 341 | Fig. 9.2: The probability that the angle between a fixed data point u with a unit-normed query q that is drawn from a spherically-symmetric distribution is at most θ, is equal to the surface area of the spherical cap with base radius a = sin θ. This fact is used in the proof of Theorem 9.1.
can be decomposed into two components: one that is parallel to the data point, râ¥(u, Ëu), and another that is orthogonal to it, râ¥(u, Ëu), as depicted in Figure 9.1. More concretely:
râ¥(u, Ëu) = â¨u â Ëu, uâ© â¥uâ¥2 u,
and,
râ¥(u, Ëu) = r(u, Ëu) â râ¥(u, Ëu).
Guo et al. [2020] show first that, regardless of the choice of Ï, the loss defined by â(u, Ëu, Ï) in Equation (9.2) can be decomposed as stated in the following theorem. | 2401.09350#341 | 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": 341,
"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. 9.2: The probability that the angle between a fixed data point u with a unit-normed query q that is drawn from a spherically-symmetric distribution is at most θ, is equal to the surface area of the spherical cap with base radius a = sin θ. This fact is used in the proof of Theorem 9.1.\ncan be decomposed into two components: one that is parallel to the data point, râ¥(u, Ëu), and another that is orthogonal to it, râ¥(u, Ëu), as depicted in Figure 9.1. More concretely:\nrâ¥(u, Ëu) = â¨u â Ëu, uâ© â¥uâ¥2 u,\nand,\nrâ¥(u, Ëu) = r(u, Ëu) â râ¥(u, Ëu).\nGuo et al. [2020] show first that, regardless of the choice of Ï, the loss defined by â(u, Ëu, Ï) in Equation (9.2) can be decomposed as stated in the following theorem.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 342 | Theorem 9.1 Given a data point u, its approximation Ëu, and any weight function Ï, the objective of Equation (9.2) can be decomposed as follows for a spherically-symmetric query distribution:
â(u, Ëu, Ï) â hâ¥(Ï, â¥uâ¥) â¥râ¥(u, Ëu)â¥2 + hâ¥(Ï, â¥uâ¥) â¥râ¥(u, Ëu)â¥2,
where,
hy(w,t) = [ w(t cos 6) ( sinâ? 6 â sin@ 6) dé,
and,
7 1 hi (w,t) = qi w(tcos 6) sin? 6 dd. â1 Jo
Proof. Without loss of generality, we can assume that queries are unit vectors (i.e., â¥q⥠= 1). Let us write â(u, Ëu, Ï) as follows:
137
138
9 Quantization
~ _ ~ (u,t,w) = B [w((a,u)) (au â a?) q = [eu cos) [fou a)2|(4,u) = [ul 054] dP [By < 0), 0 q | 2401.09350#342 | 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": 342,
"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 9.1 Given a data point u, its approximation Ëu, and any weight function Ï, the objective of Equation (9.2) can be decomposed as follows for a spherically-symmetric query distribution:\nâ(u, Ëu, Ï) â hâ¥(Ï, â¥uâ¥) â¥râ¥(u, Ëu)â¥2 + hâ¥(Ï, â¥uâ¥) â¥râ¥(u, Ëu)â¥2,\nwhere,\nhy(w,t) = [ w(t cos 6) ( sinâ? 6 â sin@ 6) dé,\nand,\n7 1 hi (w,t) = qi w(tcos 6) sin? 6 dd. â1 Jo\nProof. Without loss of generality, we can assume that queries are unit vectors (i.e., â¥q⥠= 1). Let us write â(u, Ëu, Ï) as follows:\n137\n138\n9 Quantization\n~ _ ~\u0002 (u,t,w) = B [w((a,u)) (au â a?) q = [eu cos) [fou a)2|(4,u) = [ul 054] dP [By < 0), 0 q",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 343 | where θq,u denotes the angle between q and u. θq,u ⤠θ
# Observe that P
is the surface area of a spherical cap with base radius a = â¥q⥠sin θ = sin θâsee Figure 9.2. That quantity is equal to:
â¥qâ¥dâ1 Ïd/2 Î (d/2) I(a2; d â 1 2 , 1 2 ),
where Πis the Gamma function and I(z; ·, ·) is the incomplete Beta function. We may therefore write:
ak [a s 4| x [aa _ @)3 (2) da d6 dé a dâ3 == (2 sin 6 cos 0) cos x sin?-? 6,
where in the first step we used the fact that dI(z; s, t) = (1 â z)tâ1zsâ1dz. Putting everything together, we can rewrite the loss as follows:
L(u, tw) x [ w(||u|| cos 0) E [(a. uât)?| (q,u) = |ful| cos 6] sinâ? 6 dé. 0 q
We can complete the proof by applying the following lemma to the expecta- tion over queries in the integral above.
# Lemma 9.1 | 2401.09350#343 | 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": 343,
"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": "where θq,u denotes the angle between q and u. θq,u ⤠θ\n# Observe that P\nis the surface area of a spherical cap with base radius a = â¥q⥠sin θ = sin θâsee Figure 9.2. That quantity is equal to:\nâ¥qâ¥dâ1 Ïd/2 Î (d/2) I(a2; d â 1 2 , 1 2 ),\nwhere Î is the Gamma function and I(z; ·, ·) is the incomplete Beta function. We may therefore write:\nak [a s 4| x [aa _ @)3 (2) da d6 dé a dâ3 == (2 sin 6 cos 0) cos x sin?-? 6,\nwhere in the first step we used the fact that dI(z; s, t) = (1 â z)tâ1zsâ1dz. Putting everything together, we can rewrite the loss as follows:\nL(u, tw) x [ w(||u|| cos 0) E [(a. uât)?| (q,u) = |ful| cos 6] sinâ? 6 dé. 0 q\nWe can complete the proof by applying the following lemma to the expecta- tion over queries in the integral above.\n# Lemma 9.1",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 344 | We can complete the proof by applying the following lemma to the expecta- tion over queries in the integral above.
# Lemma 9.1
2 _p 2 Bllauâa)?|(a.0) = = anu ap? + RAE Ge (ap
Proof. We use the shorthand r⥠= râ¥(u, Ëu) and similarly r⥠= râ¥(u, Ëu). Decompose q = q⥠+ q⥠where q⥠= â¨q, uâ© u â¥uâ¥2 and q⥠= q â qâ¥. We can now write:
B[(ag.u â a)°|(q.u) = #] = Bl(qy-ry)*I|(a-u) = 4] + Bllaars)I\ (au) = 4h
All other terms are equal to 0 either due to orthogonality or components or because of spherical symmetry. The first term is simply equal to â¥râ¥â¥2 t2 â¥uâ¥2 . By spherical symmetry, it is easy to show that the second term reduces to 1ât2/â¥uâ¥2 ââ dâ1
9.4 Quantization for Inner Product
Applying the lemma above to the integral, we obtain: | 2401.09350#344 | 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": 344,
"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 can complete the proof by applying the following lemma to the expecta- tion over queries in the integral above.\n# Lemma 9.1\n2 _p 2 Bllauâa)?|(a.0) = = anu ap? + RAE Ge (ap\nProof. We use the shorthand r⥠= râ¥(u, Ëu) and similarly r⥠= râ¥(u, Ëu). Decompose q = q⥠+ q⥠where q⥠= â¨q, uâ© u â¥uâ¥2 and q⥠= q â qâ¥. We can now write:\nB[(ag.u â a)°|(q.u) = #] = Bl(qy-ry)*I|(a-u) = 4] + Bllaars)I\\ (au) = 4h\nAll other terms are equal to 0 either due to orthogonality or components or because of spherical symmetry. The first term is simply equal to â¥râ¥â¥2 t2 â¥uâ¥2 . By spherical symmetry, it is easy to show that the second term reduces to 1ât2/â¥uâ¥2 ââ dâ1\n9.4 Quantization for Inner Product\nApplying the lemma above to the integral, we obtain:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 345 | 9.4 Quantization for Inner Product
Applying the lemma above to the integral, we obtain:
sin? 0 d-1 C(u, tw) | w(||ul] cos 0)( on? A||ry (uw, @)|/?+ irs) sinâ? 6 dé, 0
as desired.
When Ï(s) = 1sâ¥Î¸ for some θ, Guo et al. [2020] show that h⥠outweighs hâ¥, as the following theorem states. This implies that such an Ï puts more emphasis on preserving the parallel residual error as discussed earlier. Theorem 9.2 For Ï(s) = 1sâ¥Î¸ with θ ⥠0, hâ¥(Ï, t) ⥠hâ¥(Ï, t), with equality if and only if Ï is constant over the interval [ât, t].
Proof. We can safely assume that h⥠and h⥠are positive; they are 0 if and only if Ï(s) = 0 over [ât, t]. We can thus express the ratio between them as follows: | 2401.09350#345 | 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": 345,
"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": "9.4 Quantization for Inner Product\nApplying the lemma above to the integral, we obtain:\nsin? 0 d-1 C(u, tw) | w(||ul] cos 0)( on? A||ry (uw, @)|/?+ irs) sinâ? 6 dé, 0\nas desired.\nWhen Ï(s) = 1sâ¥Î¸ for some θ, Guo et al. [2020] show that h⥠outweighs hâ¥, as the following theorem states. This implies that such an Ï puts more emphasis on preserving the parallel residual error as discussed earlier. Theorem 9.2 For Ï(s) = 1sâ¥Î¸ with θ ⥠0, hâ¥(Ï, t) ⥠hâ¥(Ï, t), with equality if and only if Ï is constant over the interval [ât, t].\nProof. We can safely assume that h⥠and h⥠are positive; they are 0 if and only if Ï(s) = 0 over [ât, t]. We can thus express the ratio between them as follows:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 346 | hy (w, t) _ Jo w(t cos @) sin*? 6 dé yy (ta-2 | h1(w,t) (d of Je w(tcos 6) sinâ 6 dé ' (d H( Ia 1),
where we denoted by Ig = Io
0 Ï(t cos θ) sind θ dθ. Using integration by parts:
where we denoted by Ig = Io w(tcos 4) sinâ @ d@. Using integration by parts:
Tq = âw(tcos 0) cos @ sin! 6) + [ cos 6 G cos @)(d â 1) sinâ? 6 cos 6 â w(t cos @)t sin? 6] dé =(d-1) [ vtteose) cosâ @ sinâ? 6 do â t [ a'(eeos0) cos 6 sin? @ dO = (dâ 1)Ig_2 â (d-1)Ia â â[ w'(tcos 0) cos @ sinâ 6 dO. | 2401.09350#346 | 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": 346,
"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": "hy (w, t) _ Jo w(t cos @) sin*? 6 dé yy (ta-2 | h1(w,t) (d of Je w(tcos 6) sinâ 6 dé ' (d H( Ia 1),\nwhere we denoted by Ig = Io\n0 Ï(t cos θ) sind θ dθ. Using integration by parts:\nwhere we denoted by Ig = Io w(tcos 4) sinâ @ d@. Using integration by parts:\nTq = âw(tcos 0) cos @ sin! 6) + [ cos 6 G cos @)(d â 1) sinâ? 6 cos 6 â w(t cos @)t sin? 6] dé =(d-1) [ vtteose) cosâ @ sinâ? 6 do â t [ a'(eeos0) cos 6 sin? @ dO = (dâ 1)Ig_2 â (d-1)Ia â â[ w'(tcos 0) cos @ sinâ 6 dO.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 347 | Because Ï(s) = 0 for s < 0, the last term reduces to an integral over [0, Ï/2]. The resulting integral is non-negative because sine and cosine are both non- negative over that interval. It is 0 if and only if Ïâ² = 0, or equivalently when Ï is constant. We have therefore shown that:
Ta-2 hy (w,t) Ty < (d=1)la-2 â (dâ IL. d-1)(-1) 21 >I, a < (dâ1)Ia-2 â (d- 1) (1 1)(â- 2 hiqw.t) =P
with equality when Ï is constant, as desired.
# 9.4.1.2 Learning a Codebook
The results above formalize the intuition that the parallel residual plays a more important role in quantization for MIPS. If we were to plug the for139
ââ
ââ
140
9 Quantization
malism above into the objective in Equation (9.2) and optimize it to learn a codebook, we would need to compute h⥠and h⥠using Theorem 9.1. That would prove cumbersome indeed. | 2401.09350#347 | 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": 347,
"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": "Because Ï(s) = 0 for s < 0, the last term reduces to an integral over [0, Ï/2]. The resulting integral is non-negative because sine and cosine are both non- negative over that interval. It is 0 if and only if Ïâ² = 0, or equivalently when Ï is constant. We have therefore shown that:\nTa-2 hy (w,t) Ty < (d=1)la-2 â (dâ IL. d-1)(-1) 21 >I, a < (dâ1)Ia-2 â (d- 1) (1 1)(â- 2 hiqw.t) =P\nwith equality when Ï is constant, as desired.\n# 9.4.1.2 Learning a Codebook\nThe results above formalize the intuition that the parallel residual plays a more important role in quantization for MIPS. If we were to plug the for139\nââ\nââ\n140\n9 Quantization\nmalism above into the objective in Equation (9.2) and optimize it to learn a codebook, we would need to compute h⥠and h⥠using Theorem 9.1. That would prove cumbersome indeed.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 348 | Instead, Guo et al. [2020] show that Ï(s) = 1sâ¥Î¸ results in a more computationally-efficient optimization problem. Letting η(t) = hâ¥(Ï,t) hâ¥(Ï,t) , they show that η/(d â 1) concentrates around (θ/t)2 1â(θ/t)2 as d becomes larger. So in high dimensions, one can rewrite the objective function of Equation (9.2) as follows:
_ Of rally? ri(u.a)||? r) (u.a)|\? DT Grape IP + rn aIP.
Guo et al. [2020] present an optimization procedure that is based on Lloydâs iterative algorithm for KMeans, and use it to learn a codebook by minimizing the objective above. Empirically, such a codebook outperforms the one that is learnt by optimizing the reconstruction error.
# 9.4.1.3 Extensions | 2401.09350#348 | 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": 348,
"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": "Instead, Guo et al. [2020] show that Ï(s) = 1sâ¥Î¸ results in a more computationally-efficient optimization problem. Letting η(t) = hâ¥(Ï,t) hâ¥(Ï,t) , they show that η/(d â 1) concentrates around (θ/t)2 1â(θ/t)2 as d becomes larger. So in high dimensions, one can rewrite the objective function of Equation (9.2) as follows:\n_ Of rally? ri(u.a)||? r) (u.a)|\\? DT Grape IP + rn aIP.\nGuo et al. [2020] present an optimization procedure that is based on Lloydâs iterative algorithm for KMeans, and use it to learn a codebook by minimizing the objective above. Empirically, such a codebook outperforms the one that is learnt by optimizing the reconstruction error.\n# 9.4.1.3 Extensions",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 349 | # 9.4.1.3 Extensions
The score-aware quantization loss has, since its publication, been extended in two different ways. Zhang et al. [2022] adapted the objective function to an Additive Quantization form. Zhang et al. [2023] updated the weight function Ï(·) so that the importance of a data point can be estimated based on a given set of training queries. Both extensions lead to substantial improvements on benchmark datasets.
# References
F. Andr´e, A.-M. Kermarrec, and N. Le Scouarnec. Cache locality is not enough: High-performance nearest neighbor search with product quanti- zation fast scan. Proceedings of the VLDB Endowment, 9(4):288â299, 12 2015.
F. Andre, A.-M. Kermarrec, and N. Le Scouarnec. Quicker adc: Unlocking the hidden potential of product quantization with simd. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(5):1666â1677, 5 2021. A. Babenko and V. Lempitsky. The inverted multi-index. In 2012 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3069â3076, 2012. | 2401.09350#349 | 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": 349,
"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": "# 9.4.1.3 Extensions\nThe score-aware quantization loss has, since its publication, been extended in two different ways. Zhang et al. [2022] adapted the objective function to an Additive Quantization form. Zhang et al. [2023] updated the weight function Ï(·) so that the importance of a data point can be estimated based on a given set of training queries. Both extensions lead to substantial improvements on benchmark datasets.\n# References\nF. Andr´e, A.-M. Kermarrec, and N. Le Scouarnec. Cache locality is not enough: High-performance nearest neighbor search with product quanti- zation fast scan. Proceedings of the VLDB Endowment, 9(4):288â299, 12 2015.\nF. Andre, A.-M. Kermarrec, and N. Le Scouarnec. Quicker adc: Unlocking the hidden potential of product quantization with simd. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(5):1666â1677, 5 2021. A. Babenko and V. Lempitsky. The inverted multi-index. In 2012 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3069â3076, 2012.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 350 | A. Babenko and V. Lempitsky. Additive quantization for extreme vector compression. In 2014 IEEE Conference on Computer Vision and Pattern Recognition, pages 931â938, 2014.
# References
References
T. Chen, L. Li, and Y. Sun. Differentiable product quantization for end- to-end embedding compression. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 1617â1626, 7 2020.
Y. Chen, T. Guan, and C. Wang. Approximate nearest neighbor search by residual vector quantization. Sensors, 10(12):11259â11273, 2010.
T. Ge, K. He, Q. Ke, and J. Sun. Optimized product quantization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(4):744â755, 2014.
R. Gray and D. Neuhoff. Quantization. IEEE Transactions on Information Theory, 44(6):2325â2383, 1998.
R. Guo, S. Kumar, K. Choromanski, and D. Simcha. Quantization based fast inner product search. In Proceedings of the 19th International Confer- ence on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine Learning Research, pages 482â490, Cadiz, Spain, 5 2016. | 2401.09350#350 | 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": 350,
"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. Babenko and V. Lempitsky. Additive quantization for extreme vector compression. In 2014 IEEE Conference on Computer Vision and Pattern Recognition, pages 931â938, 2014.\n# References\nReferences\nT. Chen, L. Li, and Y. Sun. Differentiable product quantization for end- to-end embedding compression. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 1617â1626, 7 2020.\nY. Chen, T. Guan, and C. Wang. Approximate nearest neighbor search by residual vector quantization. Sensors, 10(12):11259â11273, 2010.\nT. Ge, K. He, Q. Ke, and J. Sun. Optimized product quantization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(4):744â755, 2014.\nR. Gray and D. Neuhoff. Quantization. IEEE Transactions on Information Theory, 44(6):2325â2383, 1998.\nR. Guo, S. Kumar, K. Choromanski, and D. Simcha. Quantization based fast inner product search. In Proceedings of the 19th International Confer- ence on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine Learning Research, pages 482â490, Cadiz, Spain, 5 2016.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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