Title: Towards dense volumetric pancreas segmentation in CT using 3D fully convolutional networks
Abstract: Pancreas segmentation in computed tomography imaging has been historically difficult for automated methods because of the large shape and size variations between patients. In this work, we describe a custom-build 3D fully convolutional network (FCN) that can process a 3D image including the whole pancreas and produce an automatic segmentation. We investigate two variations of the 3D FCN architecture; one with concatenation and one with summation skip connections to the decoder part of the network. We evaluate our methods on a dataset from a clinical trial with gastric cancer patients, including 147 contrast enhanced abdominal CT scans acquired in the portal venous phase. Using the summation architecture, we achieve an average Dice score of 89.7 $\pm$ 3.8 (range [79.8, 94.8]) % in testing, achieving the new state-of-the-art performance in pancreas segmentation on this dataset.


Title: Deformed Hamiltonian vector fields and Lagrangian fibrations
Abstract: Certain dissipative physical systems closely resemble Hamiltonian systems in $\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in each conjugate pair rescaled by a real parameter. To generalise these dynamical systems to symplectic manifolds in this paper we introduce and study the properties of deformed Hamiltonian vector fields on Lagrangian fibrations. We describe why these objects have some interesting applications to symplectic geometry and discuss how their physical interpretation motivates new problems in mathematics.


Title: Homology spheres and property R
Abstract: We present infinitely many homology spheres which contain two distinct knots whose 0-surgeries are $S^1 \times S^2$. This resolves a question posed by Kirby and Melvin in 1978.


Title: Constructing Thick $B_h$-sets
Abstract: A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\dots+a_h = b_1 + \cdots +b_h$ (with $a_i,b_i \in A$) are the trivial solutions $\{a_1,\dots,a_h\} = \{b_1,\dots,b_h\}$ (as multisets). With $h=2$ and $X={\mathbb Z}$, these sets are also known as Sidon sets, Golomb Rulers, and Babcock sets. In this work, we generalize constructions of Bose-Chowla and Singer and give the resultant bounds on the diameter of a $k$ element $B_h$ set in $\mathbb Z$ for small $k$. We conclude with a list of open problems.


Title: Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators
Abstract: Representation of nonlinear dynamical systems as infinite-dimensional linear operators over Hilbert spaces enables analysis of nonlinear systems via pseudo-spectral operator analysis. In this paper, we provide a novel representation for discrete-time control-affine nonlinear dynamical systems as linear operators acting on a Hilbert space. We also demonstrate that this representation can be used to predict the behavior of the closed-loop system given a known feedback law using recorded snapshots of the system state resulting from arbitrary, potentially open-loop control inputs. We thereby extend the predictive capabilities of dynamic mode decomposition to discrete-time nonlinear systems that are affine in control. We validate the method using two numerical experiments by predicting the response of a controlled Duffing oscillator to a known feedback law, as well as demonstrating the advantage of the developed method relative to existing techniques in the literature.


Title: Scale invariant elliptic operators with singular coefficients
Abstract: We show that a realization of the operator $L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}$ generates a semigroup in $L^p(\mathbb {R}^N)$ if and only if $D_c=b+(N-2+c)^2/4 > 0$ and $s_1+\min\{0,2-\alpha\}<N/p<s_2+\max\{0,2-\alpha\}$, where $s_i$ are the roots of the equation $b+s(N-2+c-s)=0$, or $D_c=0$ and $s_0+\min\{0,2-\alpha\} \le N/p \le s_0+\max\{0,2-\alpha\}$, where $s_0$ is the unique root of the above equation. The domain of the generator is also characterized.


Title: An extremal problem for the convolution of logarithmically concave functions
Abstract: A new position is introduced and studied for the convolution of log-concave functions, which may be regarded as a functional analogue of the maximum intersection position of convex bodies introduced and studied by Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our main result is a John-type theorem for the maximal intersection position of a pair of log-concave functions, including the corresponding decomposition of the identity. The main result holds under very weak assumptions on the functions; in particular, the functions considered may both have unbounded supports.


Title: A Universal Lipschitz Extension Property of Gromov Hyperbolic Spaces
Abstract: A metric space has the universal Lipschitz extension property if for each subspace S embedded quasi-isometrically into an arbitrary metric space M there exists a continuous linear extension of Banach-valued Lipschitz functions on S to those on all of M. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition.


Title: On the $k$-measure of partitions and distinct partitions
Abstract: The $k$-measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counting both the length and the $k$-measure for partitions and distinct partitions, respectively. The $2$-measure case for partitions extends a result of Andrews, Bhattacharjee and Dastidar.


Title: Directed Homology and Persistence Modules
Abstract: In this note, we give a self-contained account on a construction for a directed homology theory based on modules over algebras, linking it to both persistence homology and natural homology. We study its first properties, among which some exact sequences.


Title: On the total disconnectedness of the quotient Aubry set
Abstract: In this paper we show that the quotient Aubry set associated to certain Lagrangians is totally disconnected (i.e., every connected component consists of a single point). Moreover, we discuss the relation between this problem and a Morse-Sard type property for (difference of) critical subsolutions of Hamilton-Jacobi equations.


Title: Maximal Sections of Sheaves of Data over an Abstract Simplicial Complex
Abstract: We employ techniques from topological data analysis to model sensor networks. Our approach to sensor integration uses the topological method of sheaves over cell complexes. The internal consistency of data from individual sensors is determined by a set of consistency functions assigned to elements of the complex. Using these functions we determine, for any collection of data, the unique set of maximal sections of consistent data received from the sensors. We offer a proof for the existence and uniqueness of these sections and illustrate the ideas with examples.


Title: Microlocal approach to Lusztig's symmetries
Abstract: We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves.


Title: Starling: An I/O-Efficient Disk-Resident Graph Index Framework for High-Dimensional Vector Similarity Search on Data Segment
Abstract: High-dimensional vector similarity search (HVSS) is receiving a spotlight as a powerful tool for various data science and AI applications. As vector data grows larger, in-memory indexes become extremely expensive because they necessitate substantial expansion of main memory resources. One possible solution is to use disk-based implementation, which stores and searches vector data in high-performance devices like NVMe SSDs. However, HVSS for data segments is still challenging in vector databases, where one machine has multiple segments for system features (like scaling) purposes. In this setting, each segment has limited memory and disk space, so HVSS on the data segment needs to balance accuracy, efficiency, and space cost. Existing disk-based methods are sub-optimal because they do not consider all these requirements together. In this paper, we present Starling, an I/O-efficient disk-resident graph index framework that optimizes data layout and search strategy in the segment. It has two main components: (1) a data layout that includes an in-memory navigation graph and a reordered disk-based graph with locality enhancement, which reduces the search path length and disk bandwidth wastage; and (2) a block search strategy that minimizes expensive disk I/Os when executing a vector query. We conduct extensive experiments to verify Starling's effectiveness, efficiency, and scalability. On a data segment with 2GB memory and 10GB disk capacity, Starling can maintain up to 33 million vectors in 128 dimensions, and serve HVSS with more than 0.9 average precision and top-10 recall rate, and latency of under 1 millisecond. The results show that Starling exhibits 43.9$\times$ higher throughput with 98% lower query latency than state-of-the-art methods under the same accuracy.


Title: Non-singular actions of infinite-dimensional groups and polymorphisms
Abstract: Let $Z$ be a probabilistic measure space with a measure $\zeta$, $\mathbb{R}^\times$ be the multiplicative group of positive reals, let $t$ be the coordinate on $\mathbb{R}^\times$. A polymorphism of $Z$ is a measure $\pi$ on $Z\times Z\times \mathbb{R}^\times$ such that for any measurable $A$, $B\subset Z$ we have $\pi(A\times Z\times \mathbb{R}^\times)=\zeta(A)$ and the integral $\int t\,d\pi(z,u,t)$ over $Z\times B\times \mathbb{R}^\times$ is $\zeta(B)$. The set of all polymorphisms has a natural semigroup structure, the group of all nonsingular transformations is dense in this semigroup. We discuss a problem of closure in polymorphisms for certain types of infinite dimensional ('large') groups and show that a non-singular action of an infinite-dimensional group generates a representation of its train (category of double cosets) by polymorphisms.


Title: Order of uniform approximation by polynomial interpolation in the complex plane and beyond
Abstract: For Lagrange polynomial interpolation on open arcs $X=\gamma$ in $\mathbb{C}$, it is well-known that the Lebesgue constant for the family of Chebyshev points ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ on $[-1,1]\subset \mathbb{R}$ has growth order of $O(log(n))$. The same growth order was shown in [45] for the Lebesgue constant of the family ${\bf {z^{**}_n}}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of some properly adjusted Fej\'er points on a rectifiable smooth open arc $\gamma\subset \mathbb{C}$. On the other hand, in our recent work [15], it was observed that if the smooth open arc $\gamma$ is replaced by an $L$-shape arc $\gamma_0 \subset \mathbb{C}$ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fej\'er points ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ on $\gamma$, and that the rate of growth for the corresponding Lebesgue constant $L_{{\bf {z}}^{*}_n}$ is as fast as $c\,log^2(n)$ for some constant $c>0$. The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the $L$-shape arc $\gamma_0$ consisting of two line segments of the same length that meet at the angle of $\pi/2$, the growth rate of the Lebesgue constant $L_{{\bf {z}}_n^{*}}$ is at least as fast as $O(Log^2(n))$, with $\lim\sup \frac{L_{{\bf {z}}_n^{*}}}{log^2(n)} = \infty$; secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail to hold; and thirdly, a proper adjustment ${\bf z}_n^{**}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of the Fej\'er points on $\gamma$ will be described to assure the growth rate of $L_{{\bf z}_n^{**}}$ to be exactly $O(Log^2(n))$.


Title: Hamiltonicity of Schrijver graphs and stable Kneser graphs
Abstract: For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers $k\geq 1$, $s\geq 2$, and $n \geq sk+1$, the $s$-stable Kneser graph $S(n,k,s)$ has as vertices all $k$-element subsets of $[n]$ in which any two elements are in cyclical distance at least $s$. We prove that all the graphs $S(n,k,s)$, in particular Schrijver graphs $S(n,k)=S(n,k,2)$, admit a Hamilton cycle that can be computed in time $\mathcal{O}(n)$ per generated vertex.


Title: Matchings in hypercubes extend to long cycles
Abstract: The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be extended to a Hamilton cycle, i.e., to a cycle that visits every vertex exactly once. We prove that every matching of $Q_d$, $d\ge 2$, can be extended to a cycle that visits at least a $2/3$-fraction of all vertices.


Title: Adaptive Belief Discretization for POMDP Planning
Abstract: Partially Observable Markov Decision Processes (POMDP) is a widely used model to represent the interaction of an environment and an agent, under state uncertainty. Since the agent does not observe the environment state, its uncertainty is typically represented through a probabilistic belief. While the set of possible beliefs is infinite, making exact planning intractable, the belief space's complexity (and hence planning complexity) is characterized by its covering number. Many POMDP solvers uniformly discretize the belief space and give the planning error in terms of the (typically unknown) covering number. We instead propose an adaptive belief discretization scheme, and give its associated planning error. We furthermore characterize the covering number with respect to the POMDP parameters. This allows us to specify the exact memory requirements on the planner, needed to bound the value function error. We then propose a novel, computationally efficient solver using this scheme. We demonstrate that our algorithm is highly competitive with the state of the art in a variety of scenarios.


Title: Use of Jordan forms for convection-pressure split Euler solvers
Abstract: In this study, we analyze convection-pressure split Euler flux functions which contain genuine weakly hyperbolic convection subsystems. A system is said to be a genuine weakly hyperbolic if all eigenvalues are real with no complete set of linearly independent (LI) eigenvectors. To construct an upwind solver based on flux difference splitting (FDS) framework, we require to generate complete set of LI eigenvectors. This can be done through addition of generalized eigenvectors which can be computed from theory of Jordan canonical forms. Once we have complete set of LI generalized eigenvectors, we construct upwind solvers in convection-pressure splitting framework. Since generalized eigenvectors are not unique, we take extra care to ensure no direct contribution of generalized eigenvectors in the final formulation of both the newly developed numerical schemes. First scheme is based on Zha and Bilgen type splitting approach, while second is based on Toro and V\'azquez splitting. Both the schemes are tested on several bench-mark test problems on 1-D and one of them is tested on some typical 2-D test problems which involve shock instabilities. The concept of generalized eigenvector based on Jordan forms is found to be useful in dealing with the genuine weakly hyperbolic parts of the considered Euler systems.


Title: The Alexander module of a trigonal curve
Abstract: We describe the Alexander modules and Alexander polynomials (both over $\Q$ and over finite fields $\FF{p}$) of generalized trigonal curves. The rational case is closed completely; in the case of characteristic $p>0$, a few points remain open.


Title: The Zilber-Pink Conjecture and the Generalized Cosmetic Surgery Conjecture
Abstract: In this paper, we generalize the Cosmetic Surgery Conjecture to an $n$-cusped hyperbolic $3$-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For $n=1$ and $2$, we show them without the assumption.


Title: Stochastic Coalitional Games for Cooperative Random Access in M2M Communications
Abstract: In this paper, the problem of random access contention between machine type devices (MTDs) in the uplink of a wireless cellular network is studied. In particular, the possibility of forming cooperative groups to coordinate the MTDs' requests for the random access channel (RACH) is analyzed. The problem is formulated as a stochastic coalition formation game in which the MTDs are the players that seek to form cooperative coalitions to optimize a utility function that captures each MTD's energy consumption and time-varying queue length. Within each coalition, an MTD acts as a coalition head that sends the access requests of the coalition members over the RACH. One key feature of this game is its ability to cope with stochastic environments in which the arrival requests of MTDs and the packet success rate over RACH are dynamically time-varying. The proposed stochastic coalitional is composed of multiple stages, each of which corresponds to a coalitional game in stochastic characteristic form that is played by the MTDs at each time step. To solve this game, a novel distributed coalition formation algorithm is proposed and shown to converge to a stable MTD partition. Simulation results show that, on the average, the proposed stochastic coalition formation algorithm can reduce the average fail ratio and energy consumption of up to 36% and 31% for a cluster-based distribution of MTDs, respectively, compared with a noncooperative case. Moreover, when the MTDs are more sensitive to the energy consumption (queue length), the coalitions' size will increase (decrease).


Title: Weight functions on Berkovich curves
Abstract: Let $C$ be a curve over a complete discretely valued field $K$. We give tropical descriptions of the weight function attached to a pluricanonical form on $C$ and the essential skeleton of $C$. We show that the Laplacian of the weight function equals the pluricanonical divisor on Berkovich skeleta, and we describe the essential skeleton of $C$ as a combinatorial skeleton of the Berkovich skeleton of the minimal $snc$-model. In particular, if $C$ has semi-stable reduction, then the essential skeleton coincides with the minimal skeleton. As an intermediate step, we describe the base loci of logarithmic pluricanonical line bundles on minimal $snc$-models.


Title: Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs
Abstract: A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity $n^{O(\log n)}$. In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-$3$ circuits. The prior results for the sum of constantly many set-multilinear depth-$3$ circuits were only slightly better than brute-force, i.e. exponential-time. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).


Title: On the Performance of Large-Scale Wireless Networks in the Finite Block-Length Regime
Abstract: Ultra-Reliable Low-Latency Communications have stringent delay constraints, and hence use codes with small block length (short codewords). In these cases, classical models that provide good approximations to systems with infinitely long codewords become imprecise. To remedy this, in this paper, an average coding rate expression is derived for a large scale network with short codewords using stochastic geometry and the theory of coding in the finite blocklength regime. The average coding rate and upper and lower bounds on the outage probability of the large-scale network are derived, and a tight approximation of the outage probability is presented. Then, simulations are presented to study the effect of network parameters on the average coding rate and the outage probability of the network, which demonstrate that results in the literature derived for the infinite blocklength regime overestimate the network performance, whereas the results in this paper provide a more realistic performance evaluation.


Title: Hypergraph reconstruction from noisy pairwise observations
Abstract: The network reconstruction task aims to estimate a complex system's structure from various data sources such as time series, snapshots, or interaction counts. Recent work has examined this problem in networks whose relationships involve precisely two entities-the pairwise case. Here we investigate the general problem of reconstructing a network in which higher-order interactions are also present. We study a minimal example of this problem, focusing on the case of hypergraphs with interactions between pairs and triplets of vertices, measured imperfectly and indirectly. We derive a Metropolis-Hastings-within-Gibbs algorithm for this model and use the algorithms to highlight the unique challenges that come with estimating higher-order models. We show that this approach tends to reconstruct empirical and synthetic networks more accurately than an equivalent graph model without higher-order interactions.


Title: The multicolour size-Ramsey number of powers of paths
Abstract: Given a positive integer $s$, a graph $G$ is $s$-Ramsey for a graph $H$, denoted $G\rightarrow (H)_s$, if every $s$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The $s$-colour size-Ramsey number ${\hat{r}}_s(H)$ of a graph $H$ is defined to be ${\hat{r}}_s(H)=\min\{|E(G)|\colon G\rightarrow (H)_s\}$. We prove that, for all positive integers $k$ and $s$, we have ${\hat{r}}_s(P_n^k)=O(n)$, where $P_n^k$ is the $k$th power of the $n$-vertex path $P_n$.


Title: When ideals properly extend the class of Arbault sets
Abstract: In this article we continue the investigation of generalized version of Arbault sets, that was initiated in \cite{DGT} but look at the picture from the most general point of view where ideals come into play. While Arbault sets can be naturally associated with the Frechet ideal $Fin$, in \cite{DGT} it was observed that when $Fin$ is replaced by the natural density ideal $\mathcal{I}_d$ one can obtain a strictly larger class of trigonometric thin sets containing Arbault sets. From the set theoretic point of view a natural question arises as whether one can broaden the picture and specify a class of ideals (instead of a single ideal) each of which would have the similar effect. As a natural candidate, we focus on a special class of ideals, namely, non-$snt$ ideals ($snt$ stands for ``strongly non translation invariant") which properly contains the class of translation invariant ideals ($\varsupsetneq Fin$) and happens to contain ideals generated by simple density functions as also certain non-negative regular summability matrices (but not all) which can be seen from \cite{DG6}. We consider the resulting class of $\mathcal{I}$-Arbault sets and it is observed that for each such ideal, the class of $\mathcal{I}$-Arbault sets not only properly contains the class of classical Arbault sets \cite{Ar} but also a large subfamily of $\mathbf{N}$-sets (also called ``sets of absolute convergence") \cite{Ft} while being contained in the class of weak Dirichlet sets. %In particular it properly contains the family of $\mathbf{N}_0$-sets which have been extensively used in the literature (see \cite{Ar, Ka, Ko}). Though distinct from the class of $\mathbf{N}$-sets, this happens to be a new class strictly lying between the class of Arbault sets and the class of weak Dirichlet sets.


Title: Towards Palmprint Verification On Smartphones
Abstract: With the rapid development of mobile devices, smartphones have gradually become an indispensable part of people's lives. Meanwhile, biometric authentication has been corroborated to be an effective method for establishing a person's identity with high confidence. Hence, recently, biometric technologies for smartphones have also become increasingly sophisticated and popular. But it is noteworthy that the application potential of palmprints for smartphones is seriously underestimated. Studies in the past two decades have shown that palmprints have outstanding merits in uniqueness and permanence, and have high user acceptance. However, currently, studies specializing in palmprint verification for smartphones are still quite sporadic, especially when compared to face- or fingerprint-oriented ones. In this paper, aiming to fill the aforementioned research gap, we conducted a thorough study of palmprint verification on smartphones and our contributions are twofold. First, to facilitate the study of palmprint verification on smartphones, we established an annotated palmprint dataset named MPD, which was collected by multi-brand smartphones in two separate sessions with various backgrounds and illumination conditions. As the largest dataset in this field, MPD contains 16,000 palm images collected from 200 subjects. Second, we built a DCNN-based palmprint verification system named DeepMPV+ for smartphones. In DeepMPV+, two key steps, ROI extraction and ROI matching, are both formulated as learning problems and then solved naturally by modern DCNN models. The efficiency and efficacy of DeepMPV+ have been corroborated by extensive experiments. To make our results fully reproducible, the labeled dataset and the relevant source codes have been made publicly available at https://cslinzhang.github.io/MobilePalmPrint/.


Title: Luna's fundamental lemma for diagonalizable groups
Abstract: We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly\ equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, \'etale, smooth, regular, syntomic, or lci.


Title: Evolutionary Alternating Direction Method of Multipliers for Constrained Multi-Objective Optimization with Unknown Constraints
Abstract: Constrained multi-objective optimization problems (CMOPs) pervade real-world applications in science, engineering, and design. Constraint violation has been a building block in designing evolutionary multi-objective optimization algorithms for solving constrained multi-objective optimization problems. However, in certain scenarios, constraint functions might be unknown or inadequately defined, making constraint violation unattainable and potentially misleading for conventional constrained evolutionary multi-objective optimization algorithms. To address this issue, we present the first of its kind evolutionary optimization framework, inspired by the principles of the alternating direction method of multipliers that decouples objective and constraint functions. This framework tackles CMOPs with unknown constraints by reformulating the original problem into an additive form of two subproblems, each of which is allotted a dedicated evolutionary population. Notably, these two populations operate towards complementary evolutionary directions during their optimization processes. In order to minimize discrepancy, their evolutionary directions alternate, aiding the discovery of feasible solutions. Comparative experiments conducted against five state-of-the-art constrained evolutionary multi-objective optimization algorithms, on 120 benchmark test problem instances with varying properties, as well as two real-world engineering optimization problems, demonstrate the effectiveness and superiority of our proposed framework. Its salient features include faster convergence and enhanced resilience to various Pareto front shapes.


Title: Analytic Linear Lie rack Structures on Leibniz Algebras
Abstract: A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V$, \[ x\rhd y=y+\sum_{n=1}^\infty A_{n,1}(x,\ldots,x,y), \]where $A_{n,1}:V\times\ldots\times V\Leftarrow V$ is an $n+1$-multilinear map symmetric in the $n$ first arguments. In this case, $A_{1,1}$ is exactly the left Leibniz product associated to $\rhd$. Any left Leibniz algebra $(\mathfrak{h},[\;,\;])$ has a canonical analytic linear Lie rack structure given by $x\stackrel{c}{\rhd} y=\exp(\mathrm{ad}_x)(y)$, where $\mathrm{ad}_x(y)=[x,y]$. In this paper, we show that a sequence $(A_{n,1})_{n\geq1}$ of $n+1$-multilinear maps on a vector space $V$ defines an analytic linear Lie rack structure if and only if $[\;,\;]:=A_{1,1}$ is a left Leibniz bracket, the $A_{n,1}$ are invariant for $(V,[\;,\;]:)$ and satisfy a sequence of multilinear equations. Some of these equations have a cohomological interpretation and can be solved when the zero and the 1-cohomology of the left Leibniz algebra $(V,[\;,\;])$ are trivial. On the other hand, given a left Leibniz algebra $(\mathfrak{h},[\;,\;])$, we show that there is a large class of (analytic) linear Lie rack structures on $(\mathfrak{h},[\;,\;])$ which can be built from the canonical one and invariant multilinear symmetric maps on $\mathfrak{h}$. A left Leibniz algebra on which all the analytic linear Lie rack structures are build in this way will be called rigid. We use our characterizations of analytic linear Lie rack structures to show that $\mathfrak{sl}_2(\mathbb{R})$ and $\mathfrak{so}(3)$ are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra.


Title: Characterizing Boundedness in Chase Variants
Abstract: Existential rules are a positive fragment of first-order logic that generalizes function-free Horn rules by allowing existentially quantified variables in rule heads. This family of languages has recently attracted significant interest in the context of ontology-mediated query answering. Forward chaining, also known as the chase, is a fundamental tool for computing universal models of knowledge bases, which consist of existential rules and facts. Several chase variants have been defined, which differ on the way they handle redundancies. A set of existential rules is bounded if it ensures the existence of a bound on the depth of the chase, independently from any set of facts. Deciding if a set of rules is bounded is an undecidable problem for all chase variants. Nevertheless, when computing universal models, knowing that a set of rules is bounded for some chase variant does not help much in practice if the bound remains unknown or even very large. Hence, we investigate the decidability of the k-boundedness problem, which asks whether the depth of the chase for a given set of rules is bounded by an integer k. We identify a general property which, when satisfied by a chase variant, leads to the decidability of k-boundedness. We then show that the main chase variants satisfy this property, namely the oblivious, semi-oblivious (aka Skolem), and restricted chase, as well as their breadth-first versions. This paper is under consideration for publication in Theory and Practice of Logic Programming.


Title: Accelerating Text-to-Image Editing via Cache-Enabled Sparse Diffusion Inference
Abstract: Due to the recent success of diffusion models, text-to-image generation is becoming increasingly popular and achieves a wide range of applications. Among them, text-to-image editing, or continuous text-to-image generation, attracts lots of attention and can potentially improve the quality of generated images. It's common to see that users may want to slightly edit the generated image by making minor modifications to their input textual descriptions for several rounds of diffusion inference. However, such an image editing process suffers from the low inference efficiency of many existing diffusion models even using GPU accelerators. To solve this problem, we introduce Fast Image Semantically Edit (FISEdit), a cached-enabled sparse diffusion model inference engine for efficient text-to-image editing. The key intuition behind our approach is to utilize the semantic mapping between the minor modifications on the input text and the affected regions on the output image. For each text editing step, FISEdit can automatically identify the affected image regions and utilize the cached unchanged regions' feature map to accelerate the inference process. Extensive empirical results show that FISEdit can be $3.4\times$ and $4.4\times$ faster than existing methods on NVIDIA TITAN RTX and A100 GPUs respectively, and even generates more satisfactory images.


Title: Harmonic curvature in dimension four
Abstract: We provide a step towards classifying Riemannian four-manifolds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs to one of five otherwise-familiar classes of examples. The main result consists in showing that, if such a manifold (not necessarily compact or even complete) lies outside of the five classes -- a non-vacuous assumption -- then, at all points of a dense open subset, Ric has four distinct eigenvalues, while suitable local coordinates simultaneously diagonalize Ric, the metric and, in a natural sense, also the curvature tensor. Furthermore, in a local orthonormal frame formed by Ricci eigenvectors, the connection form (or, curvature tensor) has just twelve (or, respectively, six) possibly-nonzero components, which together satisfy a specific system, not depending on the point, of homogeneous polynomial equations. A part of the classification problem is thus reduced to a question in real algebraic geometry.


Title: Some remarks on infinitesimals in MV-algebras
Abstract: Replacing $\{0\}$ by the whole ideal of infinitesimals yields a weaker notion of \emph{archimedean element} that we call \emph{quasiarchimedean}. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being \emph{quasihyperarchimedean} \mbox{MV-algebras,} which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MV-algebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the \mbox{set-theoretical} usually found in the literature.


Title: On an open problem about a class of optimal ternary cyclic codes
Abstract: Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let $C_{(1,e)}$ be a cyclic code of length $3^m-1$ over GF(3) with two nonzeros $\alpha$ and $\alpha^e$, where $\alpha$ is a generator of $GF(3^m)^*$ and e is a given integer. It is shown that $C_{(1,e)}$ is optimal with parameters $[3^m-1,3^m-1-2m,4]$ if one of the following conditions is met. 1) $m\equiv0(\mathrm{mod}~ 4)$, $m\geq 4$, and $e=3^\frac{m}{2}+5$. 2) $m\equiv2(\mathrm{mod}~ 4)$, $m\geq 6$, and $e=3^\frac{m+2}{2}+5$.


Title: A Kernel Framework to Quantify a Model's Local Predictive Uncertainty under Data Distributional Shifts
Abstract: Traditional Bayesian approaches for model uncertainty quantification rely on notoriously difficult processes of marginalization over each network parameter to estimate its probability density function (PDF). Our hypothesis is that internal layer outputs of a trained neural network contain all of the information related to both its mapping function (quantified by its weights) as well as the input data distribution. We therefore propose a framework for predictive uncertainty quantification of a trained neural network that explicitly estimates the PDF of its raw prediction space (before activation), p(y'|x,w), which we refer to as the model PDF, in a Gaussian reproducing kernel Hilbert space (RKHS). The Gaussian RKHS provides a localized density estimate of p(y'|x,w), which further enables us to utilize gradient based formulations of quantum physics to decompose the model PDF in terms of multiple local uncertainty moments that provide much greater resolution of the PDF than the central moments characterized by Bayesian methods. This provides the framework with a better ability to detect distributional shifts in test data away from the training data PDF learned by the model. We evaluate the framework against existing uncertainty quantification methods on benchmark datasets that have been corrupted using common perturbation techniques. The kernel framework is observed to provide model uncertainty estimates with much greater precision based on the ability to detect model prediction errors.


Title: Identification of the Heat Transfer Coefficient Using an Inverse Heat Conduction Model
Abstract: Inverse problems of recovering heat transfer coefficient from integral measurements are considered. The heat transfer coefficient occurs in the transmission conditions of imperfect contact type or the Robin type boundary conditions. It is representable as a finite part of the Fourier series with time dependent coefficients. The additional measurements are integrals of a solution multiplied by some weights. Existence and uniqueness of solutions in Sobolev classes are proven and the conditions on the data are sharp. These conditions include smoothness and consistency conditions on the data and additional conditions on the kernels of the integral operators used in additional measurements. The proof relies on a priori bounds and the contraction mapping principle. The existence and uniqueness theorems are local in time.


Title: Heavy Ball Neural Ordinary Differential Equations
Abstract: We propose heavy ball neural ordinary differential equations (HBNODEs), leveraging the continuous limit of the classical momentum accelerated gradient descent, to improve neural ODEs (NODEs) training and inference. HBNODEs have two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly reducing the number of function evaluations (NFEs) and improving the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. We verify the advantages of HBNODEs over NODEs on benchmark tasks, including image classification, learning complex dynamics, and sequential modeling. Our method requires remarkably fewer forward and backward NFEs, is more accurate, and learns long-term dependencies more effectively than the other ODE-based neural network models. Code is available at \url{https://github.com/hedixia/HeavyBallNODE}.


Title: Precondition and Effect Reasoning for Action Recognition
Abstract: Human action recognition has drawn a lot of attention in the recent years due to the research and application significance. Most existing works on action recognition focus on learning effective spatial-temporal features from videos, but neglect the strong causal relationship among the precondition, action and effect. Such relationships are also crucial to the accuracy of action recognition. In this paper, we propose to model the causal relationships based on the precondition and effect to improve the performance of action recognition. Specifically, a Cycle-Reasoning model is proposed to capture the causal relationships for action recognition. To this end, we annotate precondition and effect for a large-scale action dataset. Experimental results show that the proposed Cycle-Reasoning model can effectively reason about the precondition and effect and can enhance action recognition performance.


Title: Sampling projections in the uniform norm
Abstract: We show that there are sampling projections on arbitrary $n$-dimensional subspaces of $B(D)$ with at most $2n$ samples and norm of order $\sqrt{n}$, where $B(D)$ is the space of complex-valued bounded functions on a set $D$. This gives a more explicit form of the Kadets-Snobar theorem for the uniform norm and improves upon Auerbach's lemma. We discuss consequences for optimal recovery in $L_p$.


Title: Some Fibonacci-Related Sequences
Abstract: We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the Walnut theorem-prover. We also prove two new results about synchronized sequences.


Title: Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain
Abstract: We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.


Title: Global topology of the Hitchin system
Abstract: Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory.


Title: Neural Collapse for Cross-entropy Class-Imbalanced Learning with Unconstrained ReLU Feature Model
Abstract: The current paradigm of training deep neural networks for classification tasks includes minimizing the empirical risk that pushes the training loss value towards zero, even after the training error has been vanished. In this terminal phase of training, it has been observed that the last-layer features collapse to their class-means and these class-means converge to the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is termed as Neural Collapse (NC). To theoretically understand this phenomenon, recent works employ a simplified unconstrained feature model to prove that NC emerges at the global solutions of the training problem. However, when the training dataset is class-imbalanced, some NC properties will no longer be true. For example, the class-means geometry will skew away from the simplex ETF when the loss converges. In this paper, we generalize NC to imbalanced regime for cross-entropy loss under the unconstrained ReLU feature model. We prove that, while the within-class features collapse property still holds in this setting, the class-means will converge to a structure consisting of orthogonal vectors with different lengths. Furthermore, we find that the classifier weights are aligned to the scaled and centered class-means with scaling factors depend on the number of training samples of each class, which generalizes NC in the class-balanced setting. We empirically prove our results through experiments on practical architectures and dataset.


Title: Testing popularity in linear time via maximum matching
Abstract: Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: given a graph G=(V,E) with strict preferences on the neighbors of the nodes, a matching M is popular if there is no other matching M' such that the number of nodes preferring M' is more than those preferring M. This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time O(|E|). Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which are a special class of popular matchings.


Title: Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold
Abstract: For a Fano manifold, We consider the geometric quantization of the K\"ahler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the finite-dimensional analogue of the optimal degeneration for the anti-canonical polarization.


Title: The Adjoint Representation of a Higher Lie Groupoid
Abstract: We extend the standard construction of the adjoint representation of a Lie groupoid to the case of an arbitrary higher Lie groupoid. As for a Lie groupoid, the adjoint representation of a higher Lie groupoid turns out to be a representation up to homotopy which is well defined up to isomorphism. Its existence and uniqueness are immediate consequences of a more general result in the theory of simplicial vector bundles: the representation up to homotopy obtained by splitting a higher vector bundle by means of a cleavage is, to within isomorphism, independent of the choice of the cleavage.


Title: DEWP: Deep Expansion Learning for Wind Power Forecasting
Abstract: Wind is one kind of high-efficient, environmentally-friendly and cost-effective energy source. Wind power, as one of the largest renewable energy in the world, has been playing a more and more important role in supplying electricity. Though growing dramatically in recent years, the amount of generated wind power can be directly or latently affected by multiple uncertain factors, such as wind speed, wind direction, temperatures, etc. More importantly, there exist very complicated dependencies of the generated power on the latent composition of these multiple time-evolving variables, which are always ignored by existing works and thus largely hinder the prediction performances. To this end, we propose DEWP, a novel Deep Expansion learning for Wind Power forecasting framework to carefully model the complicated dependencies with adequate expressiveness. DEWP starts with a stack-by-stack architecture, where each stack is composed of (i) a variable expansion block that makes use of convolutional layers to capture dependencies among multiple variables; (ii) a time expansion block that applies Fourier series and backcast/forecast mechanism to learn temporal dependencies in sequential patterns. These two tailored blocks expand raw inputs into different latent feature spaces which can model different levels of dependencies of time-evolving sequential data. Moreover, we propose an inference block corresponding for each stack, which applies multi-head self-attentions to acquire attentive features and maps expanded latent representations into generated wind power. In addition, to make DEWP more expressive in handling deep neural architectures, we adapt doubly residue learning to process stack-by-stack outputs. Finally, we present extensive experiments in the real-world wind power forecasting application on two datasets from two different turbines to demonstrate the effectiveness of our approach.


Title: Ramsey Partial Orders from Acyclic Graphs
Abstract: We prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that class of acyclic graphs has the Ramsey property and uses the partite construction.


Title: How Descriptive are GMRES Convergence Bounds?
Abstract: GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex plane. Three popular approaches pose this approximation problem on the spectrum, the field of values, or pseudospectra of the coefficient matrix. We analyze and compare these bounds, illustrating with six examples the success and failure of each. When the matrix departs from normality due only to a low-dimensional invariant subspace, we discuss how these bounds can be adapted to exploit this structure. Since the Arnoldi process that underpins GMRES provides approximations to the pseudospectra, one can estimate the GMRES convergence bounds as an iteration proceeds.


Title: GIT-Mol: A Multi-modal Large Language Model for Molecular Science with Graph, Image, and Text
Abstract: Large language models have made significant strides in natural language processing, enabling innovative applications in molecular science by processing textual representations of molecules. However, most existing language models cannot capture the rich information with complex molecular structures or images. In this paper, we introduce GIT-Mol, a multi-modal large language model that integrates the Graph, Image, and Text information. To facilitate the integration of multi-modal molecular data, we propose GIT-Former, a novel architecture that is capable of aligning all modalities into a unified latent space. We achieve a 5%-10% accuracy increase in properties prediction and a 20.2% boost in molecule generation validity compared to the baselines. With the any-to-language molecular translation strategy, our model has the potential to perform more downstream tasks, such as compound name recognition and chemical reaction prediction.


Title: On the associativity of the addition on elliptic curves
Abstract: In this short note we give a simple elementary proof of the associativity of the addition on elliptic curves. We do this by providing an explicit formula for the sum of three points, using the explicit definition of the group structure.


Title: DCR-Consistency: Divide-Conquer-Reasoning for Consistency Evaluation and Improvement of Large Language Models
Abstract: Evaluating the quality and variability of text generated by Large Language Models (LLMs) poses a significant, yet unresolved research challenge. Traditional evaluation methods, such as ROUGE and BERTScore, which measure token similarity, often fail to capture the holistic semantic equivalence. This results in a low correlation with human judgments and intuition, which is especially problematic in high-stakes applications like healthcare and finance where reliability, safety, and robust decision-making are highly critical. This work proposes DCR, an automated framework for evaluating and improving the consistency of LLM-generated texts using a divide-conquer-reasoning approach. Unlike existing LLM-based evaluators that operate at the paragraph level, our method employs a divide-and-conquer evaluator (DCE) that breaks down the paragraph-to-paragraph comparison between two generated responses into individual sentence-to-paragraph comparisons, each evaluated based on predefined criteria. To facilitate this approach, we introduce an automatic metric converter (AMC) that translates the output from DCE into an interpretable numeric score. Beyond the consistency evaluation, we further present a reason-assisted improver (RAI) that leverages the analytical reasons with explanations identified by DCE to generate new responses aimed at reducing these inconsistencies. Through comprehensive and systematic empirical analysis, we show that our approach outperforms state-of-the-art methods by a large margin (e.g., +19.3% and +24.3% on the SummEval dataset) in evaluating the consistency of LLM generation across multiple benchmarks in semantic, factual, and summarization consistency tasks. Our approach also substantially reduces nearly 90% of output inconsistencies, showing promise for effective hallucination mitigation.


Title: The near-parabolic geometry of external rays
Abstract: Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, for example providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.


Title: Allison-Benkart-Gao functor and the free non-unital alternative algebras
Abstract: Let $k$ be a field of characteristic $0$. We introduce a pair of adjoint functors, Allison-Benkart-Gao functor $\mathcal{ABG}$ and Berman-Moody functor $\mathcal{BM}$, between the category of non-unital alternative algebras over $k$ and the category ${\text{\bf Lie}_{\text{R}}}$ of Lie algebras with appropriate $sl_3(k)$-module structures. Surprisingly, when $A$ is a non-unital alternative algebra, the Allison-Gao Lie algebra $\mathcal{ABG}(A)$ is different from the more well-known Steinberg Lie algebra $st_3(A)$. Next, let $A(D)$ be the free (non-unit) alternative algebra generated by $D$ elements and $\text{Inner} A(D)$ the inner derivation algebra of $A(D)$. A conjecture on the homology of $H_r(\mathcal{ABG}(A(D)))$ is proposed. Let $A(D)_n$(resp. $\text{Inner} A(D)_n$) be the degree $n$ component of $A(D)_n$(resp. $\text{Inner} A(D)_n$). The previous conjecture implies another conjecture on the dimensions on $A(D)_n$ and $\text{Inner} A(D)_n$. We also give some evidences to support the these conjectures.


Title: A quatum inspired neural network for geometric modeling
Abstract: By conceiving physical systems as 3D many-body point clouds, geometric graph neural networks (GNNs), such as SE(3)/E(3) equivalent GNNs, have showcased promising performance. In particular, their effective message-passing mechanics make them adept at modeling molecules and crystalline materials. However, current geometric GNNs only offer a mean-field approximation of the many-body system, encapsulated within two-body message passing, thus falling short in capturing intricate relationships within these geometric graphs. To address this limitation, tensor networks, widely employed by computational physics to handle manybody systems using high-order tensors, have been introduced. Nevertheless, integrating these tensorized networks into the message-passing framework of GNNs faces scalability and symmetry conservation (e.g., permutation and rotation) challenges. In response, we introduce an innovative equivariant Matrix Product State (MPS)-based message-passing strategy, through achieving an efficient implementation of the tensor contraction operation. Our method effectively models complex many-body relationships, suppressing mean-field approximations, and captures symmetries within geometric graphs. Importantly, it seamlessly replaces the standard message-passing and layer-aggregation modules intrinsic to geometric GNNs. We empirically validate the superior accuracy of our approach on benchmark tasks, including predicting classical Newton systems and quantum tensor Hamiltonian matrices. To our knowledge, our approach represents the inaugural utilization of parameterized geometric tensor networks.


Title: The cyclic open-closed map, u-connections and R-matrices
Abstract: This paper considers the (negative) cyclic open-closed map $\mathcal{OC}^{-}$, which maps the cyclic homology of the Fukaya category of a symplectic manifold to its $S^1$-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that $\mathcal{OC}^{-}$ intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara-Levelt-Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental-Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to $\mathcal{OC}^{-}$ in the semisimple case; we also consider the non-semisimple case.


Title: Some Aspects on Solving Transportation Problem
Abstract: In this paper, we consider a class of transportation problems which arises in sample surveys and other areas of statistics. The associated cost matrices of these transportation problems are of special structure. We observe that the optimality of North West corner solution holds for the general problem where cost component is replaced by a convex function. We revisit assignment problem and present a weighted version of K$\ddot{o}$nig-Egerv$\acute{a}$ry theorem and Hungarian method. The weighted Hungarian method proposed in the paper can be used for solving transportation problem.


Title: Re-evaluating the Memory-balanced Pipeline Parallelism: BPipe
Abstract: Pipeline parallelism is an essential technique in the training of large-scale Transformer models. However, it suffers from imbalanced memory consumption, leading to insufficient memory utilization. The BPipe technique was proposed to address this issue and has proven effective in the GPT-3 model. Nevertheless, our experiments have not yielded similar benefits for LLaMA training. Additionally, BPipe only yields negligible benefits for GPT-3 training when applying flash attention. We analyze the underlying causes of the divergent performance of BPipe on GPT-3 and LLaMA. Furthermore, we introduce a novel method to estimate the performance of BPipe.


Title: Pendant appearances and components in random graphs from structured classes
Abstract: We consider random graphs sampled uniformly from a structured class of graphs, such as the class of graphs embeddable in a given surface. We sharpen and extend earlier results on pendant appearances, concerning for example numbers of leaves; and obtain results on the asymptotic distribution of components other than the giant component, under quite general conditions.


Title: The GDPR Enforcement Fines at Glance
Abstract: The General Data Protection Regulation (GDPR) came into force in 2018. After this enforcement, many fines have already been imposed by national data protection authorities in Europe. This paper examines the individual GDPR articles referenced in the enforcement decisions, as well as predicts the amount of enforcement fines with available meta-data and text mining features extracted from the enforcement decision documents. According to the results, three articles related to the general principles, lawfulness, and information security have been the most frequently referenced ones. Although the amount of fines imposed vary across the articles referenced, these three particular articles do not stand out. Furthermore, a better statistical evidence is available with other meta-data features, including information about the particular European countries in which the enforcements were made. Accurate predictions are attainable even with simple machine learning techniques for regression analysis. Basic text mining features outperform the meta-data features in this regard. In addition to these results, the paper reflects the GDPR's enforcement against public administration obstacles in the European Union (EU), as well as discusses the use of automatic decision-making systems in judiciary.


Title: Balancing Specialization and Adaptation in a Transforming Scientific Landscape
Abstract: How scientists navigate between the need to capitalize on their prior knowledge by specializing, and the urge to adapt to evolving research opportunities? Drawing from diverse perspectives on adaptation, in particular from institutional change and cultural evolution, this paper proposes a Bayesian model of the evolution of scientists' research portfolios in response to transformations in their field. The model relies on scientific abstracts and authorship data to evaluate the influence of intellectual, social, and institutional resources on scientists' trajectories within a cohort of $2\,195$ high-energy physicists between 2000 and 2019. The reallocation of research efforts is shown to be structured by learning costs, thus enhancing the utility of the scientific capital disseminated among scientists. Two dimensions of social capital, namely ``diversity'' and ``power'', have opposite effects on the magnitude of change in scientists' research interests: while ``diversity'' disrupts and expands research interests, ``power'' stabilizes physicists' research agendas -- as does institutional stability. Social capital plays a more crucial role in shifts between cognitively distant research areas.


Title: Algebraic twists of modular forms and Hecke orbits
Abstract: We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ell-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.


Title: Unified Descriptive Intensional Logic
Abstract: UDIL (unified descriptive intensional logic) aims to be an alternative and improved version of Bealer's logic fulfilling the goal of unifying Bealer's systems T1 and T2 together with adding features to deal with definite descriptions and singular terms and their related philosophical problems (there are interesting connections to Zalta's more restricted parallel second-order version in his book \emph{Axiomatic Metaphysics}). UDIL also allows a much shorter and transparent proof of soundness, in particular with regards to a notoriously difficult preliminary lemma. UDIL stands out as being both formally and philosophically distinct from mainstream approaches to intensionality. One motivation for UDIL is to contribute to the Leibnizean goal of a formal philosophy, that is, a philosophy in which arguments and proofs are carried out entirely within a formal system.


Title: Parallel Integer Sort: Theory and Practice
Abstract: Integer sorting is a fundamental problem in computer science. This paper studies parallel integer sort both in theory and in practice. In theory, we show tighter bounds for a class of existing practical integer sort algorithms, which provides a solid theoretical foundation for their widespread usage in practice and strong performance. In practice, we design a new integer sorting algorithm, \textsf{DovetailSort}, that is theoretically-efficient and has good practical performance. In particular, \textsf{DovetailSort} overcomes a common challenge in existing parallel integer sorting algorithms, which is the difficulty of detecting and taking advantage of duplicate keys. The key insight in \textsf{DovetailSort} is to combine algorithmic ideas from both integer- and comparison-sorting algorithms. In our experiments, \textsf{DovetailSort} achieves competitive or better performance than existing state-of-the-art parallel integer and comparison sorting algorithms on various synthetic and real-world datasets.


Title: Strauss- and Lions-type results for a class of Orlicz-Sobolev spaces and applications
Abstract: The main goal this work is to prove two results like Strauss and Lions for Orlicz-Sobolev spaces. After, we use these results for study the existence of solutions for a class of quasilinear problems in $\mathbb{R}^{N}$.


Title: Linear stability of compact shrinking Ricci solitons
Abstract: In this paper, we continue to investigate the second variation of Perelman's $\nu$-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No. 3, 747-763" and the more recent work in "M. Mansour and R. Asadollah, arXiv:2104.08343" and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in "arXiv:math.DG/0404165" (see also "H.-D. Cao and C. He, J. Reine Angew. Math. 2015 (2015), no. 709, 229-246.") for positive Einstein manifolds to the compact shrinking Ricci soliton case.


Title: Bring Metric Functions into Diffusion Models
Abstract: We introduce a Cascaded Diffusion Model (Cas-DM) that improves a Denoising Diffusion Probabilistic Model (DDPM) by effectively incorporating additional metric functions in training. Metric functions such as the LPIPS loss have been proven highly effective in consistency models derived from the score matching. However, for the diffusion counterparts, the methodology and efficacy of adding extra metric functions remain unclear. One major challenge is the mismatch between the noise predicted by a DDPM at each step and the desired clean image that the metric function works well on. To address this problem, we propose Cas-DM, a network architecture that cascades two network modules to effectively apply metric functions to the diffusion model training. The first module, similar to a standard DDPM, learns to predict the added noise and is unaffected by the metric function. The second cascaded module learns to predict the clean image, thereby facilitating the metric function computation. Experiment results show that the proposed diffusion model backbone enables the effective use of the LPIPS loss, leading to state-of-the-art image quality (FID, sFID, IS) on various established benchmarks.


Title: Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation on cylindrical spaces
Abstract: We prove that the Zakharov-Kuznetsov equation on cylindrical spaces is globally well-posed below the energy norm. As is known, local well-posedness below energy space was obtained by the first author. We adapt I-method to extend the solutions globally in time. Using modified energies, we obtain the polynomial bounds on the $H^s$ growth for the global solutions.


Title: Synchronized CTL over One-Counter Automata
Abstract: We consider the model-checking problem of Synchronized Computation-Tree Logic (CTL+Sync) over One-Counter Automata (OCAs). CTL+Sync augments CTL with temporal operators that require several paths to satisfy properties in a synchronous manner, e.g., the property "all paths should eventually see $p$ at the same time". The model-checking problem for CTL+Sync over finite-state Kripke structures was shown to be in $\mathsf{P}^{\mathsf{NP}^{\mathsf{NP}}}$. OCAs are labelled transition systems equipped with a non-negative counter that can be zero-tested. Thus, they induce infinite-state systems whose computation trees are not regular. The model-checking problem for CTL over OCAs was shown to be $\mathsf{PSPACE}$-complete. We show that the model-checking problem for CTL+Sync over OCAs is decidable. However, the upper bound we give is non-elementary. We therefore proceed to study the problem for a central fragment of CTL+Sync, extending CTL with operators that require all paths to satisfy properties in a synchronous manner, and show that it is in $\mathsf{EXP}^\mathsf{NEXP}$ (and in particular in $\mathsf{EXPSPACE}$), by exhibiting a certain "segmented periodicity" in the computation trees of OCAs.


Title: Mitigating Procrastination in Crowdsourcing Via Efficient Scheduling Algorithm
Abstract: Several works related to crowdsourcing have been proposed in the direction where the task executors are to perform the tasks within the stipulated deadlines. Though the deadlines are set, it may be a practical scenario that majority of the task executors submit the tasks as late as possible. This situation where the task executors may delay their task submission is termed as procrastination in behavioural economics. In many applications, these late submission of tasks may be problematic for task requesters. In literature, how to prevent this procrastination within the deadline is not addressed in crowdsourcing scenario. However, in a bipartite graph setting one procrastination aware scheduling is proposed but balanced job distribution in different slots (also termed as schedules) is not considered there. In this paper, a procrastination aware scheduling of jobs is proliferated by proposing an (randomized) algorithm in crowdsourcing scenario (also applicable in mobile and spatial crowdsourcing). Our algorithm ensures that balancing of jobs in different schedules are maintained. Our scheme is compared with the existing algorithm through extensive simulation and in terms of balancing effect, our proposed algorithm outperforms the existing one. Analytically it is shown that our proposed algorithm maintains the balanced distribution.


Title: On wormholes in the moduli space of surfaces
Abstract: We study a certain wormholing phenomenon that takes place in the Koll\'ar--Shepherd-Barron--Alexeev (KSBA) compactification of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KSBA boundary. We state a general wormhole conjecture, and we prove it for a wide range of cases. At the end, we discuss some topological properties and open questions.


Title: TR-DETR: Task-Reciprocal Transformer for Joint Moment Retrieval and Highlight Detection
Abstract: Video moment retrieval (MR) and highlight detection (HD) based on natural language queries are two highly related tasks, which aim to obtain relevant moments within videos and highlight scores of each video clip. Recently, several methods have been devoted to building DETR-based networks to solve both MR and HD jointly. These methods simply add two separate task heads after multi-modal feature extraction and feature interaction, achieving good performance. Nevertheless, these approaches underutilize the reciprocal relationship between two tasks. In this paper, we propose a task-reciprocal transformer based on DETR (TR-DETR) that focuses on exploring the inherent reciprocity between MR and HD. Specifically, a local-global multi-modal alignment module is first built to align features from diverse modalities into a shared latent space. Subsequently, a visual feature refinement is designed to eliminate query-irrelevant information from visual features for modal interaction. Finally, a task cooperation module is constructed to refine the retrieval pipeline and the highlight score prediction process by utilizing the reciprocity between MR and HD. Comprehensive experiments on QVHighlights, Charades-STA and TVSum datasets demonstrate that TR-DETR outperforms existing state-of-the-art methods. Codes are available at \url{https://github.com/mingyao1120/TR-DETR}.


Title: Integer Forcing-and-Forward Transceiver Design for MIMO Multi-Pair Two-Way Relaying
Abstract: In this paper, we propose a new transmission scheme, named as Integer Forcing-and-Forward (IFF), for communications among multi-pair multiple-antenna users in which each pair exchanges their messages with the help of a single multi antennas relay in the multiple-access and broadcast phases. The proposed scheme utilizes Integer Forcing Linear Receiver (IFLR) at relay, which uses equations, i.e., linear integer-combinations of messages, to harness the intra-pair interference. Accordingly, we propose the design of mean squared error (MSE) based transceiver, including precoder and projection matrices for the relay and users, assuming that the perfect channel state information (CSI) is available. In this regards, in the multiple-access phase, we introduce two new MSE criteria for the related precoding and filter designs, i.e., the sum of the equations MSE (Sum-Equation MSE) and the maximum of the equations MSE (Max-Equation MSE), to exploit the equations in the relay. In addition, the convergence of the proposed criteria is proven as well. Moreover, in the broadcast phase, we use the two traditional MSE criteria, i.e. the sum of the users' mean squred errors (Sum MSE) and the maximum of the users' mean squared errors (Max MSE), to design the related precoding and filters for recovering relay's equations by the users. Then, we consider a more practical scenario with imperfect CSI. For this case, IFLR receiver is modified, and another transceiver design is proposed, which take into account the effect of channels estimation error. We evaluate the performance of our proposed strategy and compare the results with the conventional amplify-and-forward (AF) and denoise-and-forward (DF) strategies for the same scenario. The results indicate the substantial superiority of the proposed strategy in terms of the outage probability and the sum rate.


Title: Homological properties of the relative Frobenius morphism
Abstract: This work concerns maps of commutative noetherian local rings containing a field of positive characteristic. Given such a map $\varphi$ of finite flat dimension, the results relate homological properties of the relative Frobenius of $\varphi$ to those of the fibers of $\varphi$. The focus is on the complete intersection property and the Gorenstein property.


Title: A Truthful Referral Auction Over Networks
Abstract: This paper studies a mechanism design problem over a network, where agents can only participate by referrals. The Bulow-Klemberer theorem proposes that expanding the number of participants is a more effective approach to increase revenue than modifying the auction format. However, agents lack the motivation to invite others because doing so intensifies competition among them. On the other hand, misreporting social networks is also a common problem that can reduce revenue. Examples of misreporting include Sybil attacks (an agent pretending to be multiple bidders) and coalition groups (multiple agents pretending to be an agent). To address these challenges, we introduce a novel mechanism called the Truthful Referral Diffusion Mechanism (TRDM). TRDM incentivizes agents to report their social networks truthfully, and some of them are rewarded by the seller for improving revenue. In spite of the fact that some agents overbid in TRDM, the revenue is fixed, and it is higher than the revenue of any mechanism without referrals. TRDM is budget-balanced (non-negative revenue) and generates an efficient outcome (maximized social welfare), making it attractive for both the seller and the buyers as it improves revenue and reward.


Title: LLaVA-$φ$: Efficient Multi-Modal Assistant with Small Language Model
Abstract: In this paper, we introduce LLaVA-$\phi$ (LLaVA-Phi), an efficient multi-modal assistant that harnesses the power of the recently advanced small language model, Phi-2, to facilitate multi-modal dialogues. LLaVA-Phi marks a notable advancement in the realm of compact multi-modal models. It demonstrates that even smaller language models, with as few as 2.7B parameters, can effectively engage in intricate dialogues that integrate both textual and visual elements, provided they are trained with high-quality corpora. Our model delivers commendable performance on publicly available benchmarks that encompass visual comprehension, reasoning, and knowledge-based perception. Beyond its remarkable performance in multi-modal dialogue tasks, our model opens new avenues for applications in time-sensitive environments and systems that require real-time interaction, such as embodied agents. It highlights the potential of smaller language models to achieve sophisticated levels of understanding and interaction, while maintaining greater resource efficiency.The project is available at {https://github.com/zhuyiche/llava-phi}.


Title: Asymptotically Optimal Proper Conflict-Free Colouring
Abstract: A proper conflict-free colouring of a graph is a colouring of the vertices such that any two adjacent vertices receive different colours, and for every non-isolated vertex $v$, some colour appears exactly once on the neighbourhood of $v$. Caro, Petru\v{s}evski and \v{S}krekovski conjectured that every connected graph with maximum degree $\Delta \geq 3$ has a proper conflict-free colouring with at most $\Delta+1$ colours. This conjecture holds for $\Delta=3$ and remains open for $\Delta \geq 4$. In this paper we prove that this conjecture holds asymptotically; namely, every graph with maximum degree $\Delta$ has a proper conflict-free colouring with $(1+o(1))\Delta$ colours.


Title: Extreme statistics of non-intersecting Brownian paths
Abstract: We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [MFQR13] for the joint distribution of $\mathcal{M}={\rm max}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$ and $\mathcal{T}={\rm argmax}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$, where $\mathcal{A}_2$ is the Airy$_2$ process, and we use them to show that in the three cases the joint distribution converges, as $N\to\infty$, to the joint distribution of $\mathcal{M}$ and $\mathcal{T}$. In the case of non-intersecting Brownian bridges on the line, we also establish small deviation inequalities for the argmax which match the tail behavior of $\mathcal{T}$. Our proofs are based on the method introduced in [CQR13,BCR15] for obtaining formulas for the probability that the top line of these line ensembles stays below a given curve, which are given in terms of the Fredholm determinant of certain "path-integral" kernels.


Title: Convergence of Stochastic Gradient Descent for PCA
Abstract: We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i.i.d. data points in $\reals^d$. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in \cite{hardt2014noisy}. Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap.


Title: Politics and Propaganda on Social Media: How Twitter and Meta Moderate State-Linked Information Operations
Abstract: Why do Social Media Corporations (SMCs) engage in state-linked information operations? Social media can significantly influence the global political landscape, allowing governments and other political entities to engage in concerted information operations, shaping or manipulating domestic and foreign political agendas. In response to state-linked political manipulation tactics on social media, Twitter and Meta carried out take-down operations against propaganda networks, accusing them of interfering foreign elections, organizing disinformation campaigns, manipulating political debates and many other issues. This research investigates the two SMCs' policy orientation to explain which factors can affect these two companies' reaction against state-linked information operations. We find that good governance indicators such as democracy are significant elements of SMCs' country-focus. This article also examines whether Meta and Twitter's attention to political regime characteristics is influenced by international political alignments. This research illuminates recent trends in SMCs' take-down operations and illuminating interplay between geopolitics and domestic regime characteristics.


Title: Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels
Abstract: We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying $C_1 \left(x^{-a}y^{b}+x^{b}y^{-a}\right)\leq K\left(x,y\right)\leq C_2\left(x^{-a}y^{b}+x^{b}y^{-a}\right)$ with $a>0$ and $b<1$. This covers especially the case of Smoluchowski's classical kernel $K(x,y)=(x^{1/3} + y^{1/3})(x^{-1/3} + y^{-1/3})$. For the proof of existence we first consider some regularized kernel $K_{\epsilon}$ for which we construct a sequence of solutions $h_{\epsilon}$. In a second step we pass to the limit $\epsilon\to 0$ to obtain a solution for the original kernel $K$. The main difficulty is to establish a uniform lower bound on $h_{\epsilon}$. The basic idea for this is to consider the time-dependent problem and choosing a special test function that solves the dual problem.


Title: Hörmander properties of discrete time Markov processes
Abstract: We present an abstract framework for establishing smoothing properties within a specific class of inhomogeneous discrete-time Markov processes. These properties, in turn, serve as a basis for demonstrating the existence of a density function for our process or more precisely for a regularized version of it. They can also be exploited to show its total variation convergence towards the solution of a Stochastic Differential Equation as the time step between two observations of our discrete time Markov process tends to zero. The distinctive feature of our methodology lies in the exploration of smoothing properties under a local weak H\"ormander type condition satisfied by the discrete-time Markov process. Our H\"ormander property is demonstrated to align with the standard local weak H\"ormander associated to the Stochastic Differential Equation which is the total variation limit of our discrete time Markov process.


Title: ChartAssisstant: A Universal Chart Multimodal Language Model via Chart-to-Table Pre-training and Multitask Instruction Tuning
Abstract: Charts play a vital role in data visualization, understanding data patterns, and informed decision-making. However, their unique combination of graphical elements (e.g., bars, lines) and textual components (e.g., labels, legends) poses challenges for general-purpose multimodal models. While vision-language models trained on chart data excel in comprehension, they struggle with generalization and require task-specific fine-tuning. To address these challenges, we propose ChartAssistant, a chart-based vision-language model for universal chart comprehension and reasoning. ChartAssistant leverages ChartSFT, a comprehensive dataset covering diverse chart-related tasks with basic and specialized chart types. It undergoes a two-stage training process, starting with pre-training on chart-to-table parsing to align chart and text, followed by multitask instruction-following fine-tuning. This approach enables ChartAssistant to achieve competitive performance across various chart tasks without task-specific fine-tuning. Experimental results demonstrate significant performance gains over the state-of-the-art UniChart method, outperforming OpenAI's GPT-4V(ision) on real-world chart data. The code and data are available at https://github.com/OpenGVLab/ChartAst.


Title: Regular polygraphs and the Simpson conjecture
Abstract: We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two different notions of "non-unital categories". The "general" version involve infinity-categories where absolutely all composition operations (horizontal, vertical and whiskering) are defined and compatible, the "regular" version involve infinity-categories where all the composition operations corresponding to "regular" pasting diagram are defined and compatible. In both case we construct (weak) model structures on these categories such that fibrant objects have weak units and weak inverse. We prove the regular version of the conjecture using the original strategy of Kapranov and Voevodsky, together with our previous work on polygraphs. The general version cannot be proved by these methods and is still open. In order to do this we also study some subtle property of the combinatorics of polygraphs, and we construct a new counting function for polygraphs, inspired by previous work of Makkai.


Title: Derivatives of symplectic eigenvalues and a Lidskii type theorem
Abstract: Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist $n$ positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb{R}^{2n}$ called the symplectic eigenbasis of $A$ corresponding to these numbers. In this paper, we discuss the differentiability (analyticity) of the symplectic eigenvalues and corresponding symplectic eigenbasis for differentiable (analytic) map $t\mapsto A(t),$ and compute their derivatives. We then derive an analogue of Lidskii's theorem for symplectic eigenvalues as an application.


Title: Shrinking unit: a Graph Convolution-Based Unit for CNN-like 3D Point Cloud Feature Extractors
Abstract: 3D point clouds have attracted increasing attention in architecture, engineering, and construction due to their high-quality object representation and efficient acquisition methods. Consequently, many point cloud feature detection methods have been proposed in the literature to automate some workflows, such as their classification or part segmentation. Nevertheless, the performance of point cloud automated systems significantly lags behind their image counterparts. While part of this failure stems from the irregularity, unstructuredness, and disorder of point clouds, which makes the task of point cloud feature detection significantly more challenging than the image one, we argue that a lack of inspiration from the image domain might be the primary cause of such a gap. Indeed, given the overwhelming success of Convolutional Neural Networks (CNNs) in image feature detection, it seems reasonable to design their point cloud counterparts, but none of the proposed approaches closely resembles them. Specifically, even though many approaches generalise the convolution operation in point clouds, they fail to emulate the CNNs multiple-feature detection and pooling operations. For this reason, we propose a graph convolution-based unit, dubbed Shrinking unit, that can be stacked vertically and horizontally for the design of CNN-like 3D point cloud feature extractors. Given that self, local and global correlations between points in a point cloud convey crucial spatial geometric information, we also leverage them during the feature extraction process. We evaluate our proposal by designing a feature extractor model for the ModelNet-10 benchmark dataset and achieve 90.64% classification accuracy, demonstrating that our innovative idea is effective. Our code is available at github.com/albertotamajo/Shrinking-unit.


Title: On the use of the M-quantiles for outlier detection in multivariate data
Abstract: Defining a successful notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of [8] and [25] leading to M-quantiles, is very appealing for its mathematical elegance combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of R^d and the unit ball in R^d. By analogy with the d=1 case where the K-transform is a cumulative distribution function-like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d>=1. Over the past twenty years the resulting M-quantiles have seen applications in a variety of fields, primarily for the purpose of detecting outliers in multidimensional spaces. In this article we prove that for odd d>=3, it is not the gradient but a poly-Laplacian of the K-function that is (almost everywhere) proportional to the density function. For d even one cannot establish a differential equation connecting the K-function with the density. These results show that usage of the K-transform for outlier detection in higher odd-dimensions is in principle flawed, as the K-transform does not originate from inversion of a true M-distribution. We demonstrate these conclusions in two dimensions through examples from non-standard asymmetric distributions. Our examples illustrate a feature of the K-transform whereby regions in the domain with higher density map to larger volumes in the co-domain, thereby producing a magnification effect that moves inliers closer to the boundary of the co-domain than outliers. This feature obviously disrupts any outlier detection mechanism that relies on the inverse K-transform.


Title: Multi-Auxiliary Augmented Collaborative Variational Auto-encoder for Tag Recommendation
Abstract: Recommending appropriate tags to items can facilitate content organization, retrieval, consumption and other applications, where hybrid tag recommender systems have been utilized to integrate collaborative information and content information for better recommendations. In this paper, we propose a multi-auxiliary augmented collaborative variational auto-encoder (MA-CVAE) for tag recommendation, which couples item collaborative information and item multi-auxiliary information, i.e., content and social graph, by defining a generative process. Specifically, the model learns deep latent embeddings from different item auxiliary information using variational auto-encoders (VAE), which could form a generative distribution over each auxiliary information by introducing a latent variable parameterized by deep neural network. Moreover, to recommend tags for new items, item multi-auxiliary latent embeddings are utilized as a surrogate through the item decoder for predicting recommendation probabilities of each tag, where reconstruction losses are added in the training phase to constrict the generation for feedback predictions via different auxiliary embeddings. In addition, an inductive variational graph auto-encoder is designed where new item nodes could be inferred in the test phase, such that item social embeddings could be exploited for new items. Extensive experiments on MovieLens and citeulike datasets demonstrate the effectiveness of our method.


Title: Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\Ga)$
Abstract: We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into $c_0(\Ga)$ and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces $L_p(\mu)$, $1\le p\le2$) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that $X=L_p(\mu)$, $1\le p<\infty$, bi-Lipschitz embeds into $c_0(\Ga)$ if and only if $\dens X<\om_\om$.


Title: A Soft Recommender System for Social Networks
Abstract: Recent social recommender systems benefit from friendship graph to make an accurate recommendation, believing that friends in a social network have exactly the same interests and preferences. Some studies have benefited from hard clustering algorithms (such as K-means) to determine the similarity between users and consequently to define degree of friendships. In this paper, we went a step further to identify true friends for making even more realistic recommendations. we calculated the similarity between users, as well as the dependency between a user and an item. Our hypothesis is that due to the uncertainties in user preferences, the fuzzy clustering, instead of the classical hard clustering, is beneficial in accurate recommendations. We incorporated the C-means algorithm to get different membership degrees of soft users' clusters. Then, the users' similarity metric is defined according to the soft clusters. Later, in a training scheme we determined the latent representations of users and items, extracting from the huge and sparse user-item-tag matrix using matrix factorization. In the parameter tuning, we found the optimum coefficients for the influence of our soft social regularization and the user-item dependency terms. Our experimental results convinced that the proposed fuzzy similarity metric improves the recommendations in real data compared to the baseline social recommender system with the hard clustering.


Title: Energy Identity for Stationary Harmonic Maps
Abstract: In this paper we consider sequences $u_j:B_2\subseteq M\to N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j|^2\leq \Lambda$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+\nu$, where $\nu = e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(\nu)$ one can produce a finite number of bubble maps $b_j:S^2\to N$ by blowing up the sequence $u_j$ near $x$. We prove the energy identity in this paper. Namely, we have at a.e. $x\in S$ that $e(x)=\sum_j E[b_j]$ for a complete set of such bubbles. That is, the energy density of the defect measure $\nu$ is precisely the sum of the energies of the bubbling maps.


Title: InstructTA: Instruction-Tuned Targeted Attack for Large Vision-Language Models
Abstract: Large vision-language models (LVLMs) have demonstrated their incredible capability in image understanding and response generation. However, this rich visual interaction also makes LVLMs vulnerable to adversarial examples. In this paper, we formulate a novel and practical gray-box attack scenario that the adversary can only access the visual encoder of the victim LVLM, without the knowledge of its prompts (which are often proprietary for service providers and not publicly available) and its underlying large language model (LLM). This practical setting poses challenges to the cross-prompt and cross-model transferability of targeted adversarial attack, which aims to confuse the LVLM to output a response that is semantically similar to the attacker's chosen target text. To this end, we propose an instruction-tuned targeted attack (dubbed InstructTA) to deliver the targeted adversarial attack on LVLMs with high transferability. Initially, we utilize a public text-to-image generative model to "reverse" the target response into a target image, and employ GPT-4 to infer a reasonable instruction $\boldsymbol{p}^\prime$ from the target response. We then form a local surrogate model (sharing the same visual encoder with the victim LVLM) to extract instruction-aware features of an adversarial image example and the target image, and minimize the distance between these two features to optimize the adversarial example. To further improve the transferability, we augment the instruction $\boldsymbol{p}^\prime$ with instructions paraphrased from an LLM. Extensive experiments demonstrate the superiority of our proposed method in targeted attack performance and transferability.


Title: Sampling probabilities, diffusions, ancestral graphs, and duality under strong selection
Abstract: Wright-Fisher diffusions and their dual ancestral graphs occupy a central role in the study of allele frequency change and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the sampling probability, a crucial quantity in inference. Under a finite-allele mutation model, with possibly parent-dependent mutation, we consider the asymptotic regime where the selective advantage of one allele grows to infinity, while the other parameters remain fixed. In this regime, we show that the Wright-Fisher diffusion can be approximated either by a Gaussian process or by a process whose components are independent continuous-state branching processes with immigration, aligning with analogous results for Wright-Fisher models but employing different methods. While the first process becomes degenerate at stationarity, the latter does not and provides a simple, analytic approximation for the leading term of the sampling probability. Furthermore, using another approach based on a recursion formula, we characterise all remaining terms to provide a full asymptotic expansion for the sampling probability. Finally, we study the asymptotic behaviour of the rates of the block-counting process of the conditional ancestral selection graph and establish an asymptotic duality relationship between this and the diffusion.