Title: DeepOnet Based Preconditioning Strategies For Solving Parametric Linear Systems of Equations
Abstract: We introduce a new class of hybrid preconditioners for solving parametric linear systems of equations. The proposed preconditioners are constructed by hybridizing the deep operator network, namely DeepONet, with standard iterative methods. Exploiting the spectral bias, DeepONet-based components are harnessed to address low-frequency error components, while conventional iterative methods are employed to mitigate high-frequency error components. Our preconditioning framework comprises two distinct hybridization approaches: direct preconditioning (DP) and trunk basis (TB) approaches. In the DP approach, DeepONet is used to approximate an action of an inverse operator to a vector during each preconditioning step. In contrast, the TB approach extracts basis functions from the trained DeepONet to construct a map to a smaller subspace, in which the low-frequency component of the error can be effectively eliminated. Our numerical results demonstrate that utilizing the TB approach enhances the convergence of Krylov methods by a large margin compared to standard non-hybrid preconditioning strategies. Moreover, the proposed hybrid preconditioners exhibit robustness across a wide range of model parameters and problem resolutions.


Title: ETTR Bounds and Approximation Solutions of Blind Rendezvous Policies in Cognitive Radio Networks with Random Channel States
Abstract: In this paper, we consider the multichannel rendezvous problem in cognitive radio networks (CRNs) where the probability that two users hopping on the same channel have a successful rendezvous is a function of channel states. The channel states are modeled by stochastic processes with joint distributions known to users. However, the exact state of a channel at any time is not observable. We first consider two channel models: (i) the fast time-varying channel model (where the channel states are assumed to be independent and identically distributed in each time slot), and (ii) the slow time-varying channel model (where the channel states remain unchanged over time). Among the classes of the blind rendezvous policies that randomly hop on channels according to certain channel selection probabilities, we show the optimal channel selection policy that minimizes the expected time-to-rendezvous (ETTR) is the single selection policy that hops on the ``best'' channel all the time in the fast time-varying channel model. However, for the slow time-varying channel model, it is much more difficult to find the optimal channel selection policy. By using the majorization ordering, we derive a lower bound and an upper bound for the ETTR under the assumption that the channel states are exchangeable random variables. Bases on these bounds, we then prove various approximation solutions. We then extend our results to general channel models where the joint distribution of the channel states is only assumed to be stationary in time.


Title: On the intermittency front of stochastic heat equation driven by colored noises
Abstract: We study the propagation of high peaks (intermittency front) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R}^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.


Title: Relative Entropy for Quantum Channels
Abstract: We introduce an quantum entropy for bimodule quantum channels on finite von Neumann algebras, generalizing the remarkable Pimsner-Popa entropy. The relative entropy for Fourier multipliers of bimodule quantum channels establishes an upper bound of the quantum entropy. Additionally, we present the Araki relative entropy for bimodule quantum channels, revealing its equivalence to the relative entropy for Fourier multipliers and demonstrating its left/right monotonicities and convexity. Notably, the quantum entropy attains its maximum if there is a downward Jones basic construction. By considering R\'{e}nyi entropy for Fourier multipliers, we find a continuous bridge between the logarithm of the Pimsner-Popa index and the Pimsner-Popa entropy. As a consequence, the R\'{e}nyi entropy at $1/2$ serves a criterion for the existence of a downward Jones basic construction.


Title: Temporal Matrix Factorization for Tracking Concept Drift in Individual User Preferences
Abstract: The matrix factorization (MF) technique has been widely adopted for solving the rating prediction problem in recommender systems. The MF technique utilizes the latent factor model to obtain static user preferences (user latent vectors) and item characteristics (item latent vectors) based on historical rating data. However, in the real world user preferences are not static but full of dynamics. Though there are several previous works that addressed this time varying issue of user preferences, it seems (to the best of our knowledge) that none of them is specifically designed for tracking concept drift in individual user preferences. Motivated by this, we develop a Temporal Matrix Factorization approach (TMF) for tracking concept drift in each individual user latent vector. There are two key innovative steps in our approach: (i) we develop a modified stochastic gradient descent method to learn an individual user latent vector at each time step, and (ii) by the Lasso regression we learn a linear model for the transition of the individual user latent vectors. We test our method on a synthetic dataset and several real datasets. In comparison with the original MF, our experimental results show that our temporal method is able to achieve lower root mean square errors (RMSE) for both the synthetic and real datasets. One interesting finding is that the performance gain in RMSE is mostly from those users who indeed have concept drift in their user latent vectors at the time of prediction. In particular, for the synthetic dataset and the Ciao dataset, there are quite a few users with that property and the performance gains for these two datasets are roughly 20% and 5%, respectively.


Title: Quotient polynomials with positive coefficients
Abstract: We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.


Title: Tangent categories of algebras over operads
Abstract: Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper.


Title: The Optimal Paper Moebius Band
Abstract: In this paper we prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must converge, up to isometry, to the famous triangular Moebius band. These results answer the minimum aspect ratio question discussed by W. Wunderlich in 1962 and prove the more specific conjecture of B Halpern and C. Weaver from 1977.


Title: The Evolving Ecosystem of Predatory Journals: A Case Study in Indian Perspective
Abstract: Digital advancement in scholarly repositories has led to the emergence of a large number of open access predatory publishers that charge high article processing fees from authors but fail to provide necessary editorial and publishing services. Identifying and blacklisting such publishers has remained a research challenge due to the highly volatile scholarly publishing ecosystem. This paper presents a data-driven approach to study how potential predatory publishers are evolving and bypassing several regularity constraints. We empirically show the close resemblance of predatory publishers against reputed publishing groups. In addition to verifying standard constraints, we also propose distinctive signals gathered from network-centric properties to understand this evolving ecosystem better.


Title: Maximal polarization for periodic configurations on the real line
Abstract: We prove that among all 1-periodic configurations $\Gamma$ of points on the real line $\mathbb{R}$ the quantities $$ \min_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \pi \alpha (x - \gamma)^2} \quad \text{and} \quad \max_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \pi \alpha (x - \gamma)^2}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $n$ per period is sufficiently large (depending on $\alpha$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$ for large $n$. The first result is shown using Fourier series. The second result follows from work of Cohn and Kumar on universal optimality and holds for all $n$ (independent of $\alpha$).


Title: Grounding Complex Navigational Instructions Using Scene Graphs
Abstract: Training a reinforcement learning agent to carry out natural language instructions is limited by the available supervision, i.e. knowing when the instruction has been carried out. We adapt the CLEVR visual question answering dataset to generate complex natural language navigation instructions and accompanying scene graphs, yielding an environment-agnostic supervised dataset. To demonstrate the use of this data set, we map the scenes to the VizDoom environment and use the architecture in \citet{gatedattention} to train an agent to carry out these more complex language instructions.


Title: Demystifying $μ$
Abstract: We develop the theory of illfounded and cyclic proof systems in the context of the modal $\mu$-calculus. A fine analysis of provability and admissibility bridges the finitary, cyclic and illfounded notions of proof for this logic and re-enforces the subtlety of two important normal form theorems: guardedness and disjunctiveness.


Title: Identifying contact graphs of sphere packings with generic radii
Abstract: Ozkan et al. conjectured that any packing of $n$ spheres with generic radii will be stress-free, and hence will have at most $3n-6$ contacts. In this paper we prove that this conjecture is true for any sphere packing with contact graph of the form $G \oplus K_2$, i.e., the graph formed by connecting every vertex in a graph $G$ to every vertex in the complete graph with two vertices. We also prove the converse of the conjecture holds in this special case: specifically, a graph $G \oplus K_2$ is the contact graph of a generic radii sphere packing if and only if $G$ is a penny graph with no cycles.


Title: A Generalized Variable Projection Algorithm for Least Squares Problems in Atmospheric Remote Sensing
Abstract: This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the nonlinear parameters remain consistent across all datasets. A well-established approach for solving such problems is the variable projection algorithm introduced by Golub and LeVeque, which effectively reduces a separable problem to its nonlinear component. However, this algorithm assumes that the datasets have equal sizes and identical auxiliary model parameters. This article is motivated by a real-world remote sensing application where these assumptions do not apply. Consequently, we propose a generalized algorithm that extends the original theory to overcome these limitations. The new algorithm has been implemented and tested using both synthetic and real satellite data for atmospheric carbon dioxide retrievals. It has also been compared to conventional state-of-the-art solvers, and its advantages are thoroughly discussed. The experimental results demonstrate that the proposed algorithm significantly outperforms all other methods in terms of computation time, while maintaining comparable accuracy and stability. Hence, this novel method can have a positive impact on future applications in remote sensing and could be valuable for other scientific fitting problems with similar properties.


Title: Mean Value Theorems for Binary Egyptian Fractions
Abstract: In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.


Title: Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs
Abstract: The enumeration of Hamiltonian cycles on 2n*2n grids of nodes is a longstanding problem in combinatorics. Previous work has concentrated on counting all cycles. The current work enumerates nonisomorphic cycles -- that is, the number of isomorphism classes (up to all symmetry operations of the square). It is shown that the matrix method used previously can be modified to count cycles with all combinations of reflective and 180-degree rotational symmetry. Cycles with 90-degree rotational symmetry were counted by a direct search, using a modification of Knuth's Dancing Links algorithm. From these counts, the numbers of nonisomorphic cycles were calculated for n<=10.


Title: Schur Multipliers of Nilpotent Lie Algebras
Abstract: We consider the Schur multipliers of finite dimensional nilpotent Lie algebras. If the algebra has dimension greater than one, then the Schur multiplier is non-zero. We give a direct proof of an upper bound for the dimension of the Schur multiplier as a function of class and the minimum number of generators of the algebra. We then compare this bound with another known bound.


Title: Human activity recognition from mobile inertial sensors using recurrence plots
Abstract: Inertial sensors are present in most mobile devices nowadays and such devices are used by people during most of their daily activities. In this paper, we present an approach for human activity recognition based on inertial sensors by employing recurrence plots (RP) and visual descriptors. The pipeline of the proposed approach is the following: compute RPs from sensor data, compute visual features from RPs and use them in a machine learning protocol. As RPs generate texture visual patterns, we transform the problem of sensor data classification to a problem of texture classification. Experiments for classifying human activities based on accelerometer data showed that the proposed approach obtains the highest accuracies, outperforming time- and frequency-domain features directly extracted from sensor data. The best results are obtained when using RGB RPs, in which each RGB channel corresponds to the RP of an independent accelerometer axis.


Title: A Survey Analyzing Generalization in Deep Reinforcement Learning
Abstract: Reinforcement learning research obtained significant success and attention with the utilization of deep neural networks to solve problems in high dimensional state or action spaces. While deep reinforcement learning policies are currently being deployed in many different fields from medical applications to self driving vehicles, there are still ongoing questions the field is trying to answer on the generalization capabilities of deep reinforcement learning policies. In this paper, we will outline the fundamental reasons why deep reinforcement learning policies encounter overfitting problems that limit their robustness and generalization capabilities. Furthermore, we will formalize and unify the diverse solution approaches to increase generalization, and overcome overfitting in state-action value functions. We believe our study can provide a compact systematic unified analysis for the current advancements in deep reinforcement learning, and help to construct robust deep neural policies with improved generalization abilities.


Title: Hochschild and cyclic Homologies with bounded poles
Abstract: We show that the classical Hochschild homology and (periodic and negative) cyclic homology groups are representable in the category of motives with modulus. We do this by constructing Hochschild homology and (periodic and negative) cyclic homologies for modulus pairs. We show a modulus version of HKR theorem, that is, there exists an isomorphism between modulus Hochschild homology and modulus K\"ahler differentials for affine normal crossing modulus pairs. By using the representability of modulus Hodge sheaves in the category of motives with modulus, we construct an object of the category of motives with modulus which represents modulus Hochschild homology. Similarly, We compare modulus de Rham cohomology and modulus cyclic homologies and obtain a representability of modulus cyclic homologies.


Title: Travelers: A scalable fair ordering BFT system
Abstract: Many blockchain platform are subject to maximal value extraction (MEV), and users on the platform are losing money while sending transactions because the transaction order can be manipulated to extract value from them. Consensus protocols have been augmented with different notion of fair ordering in order to counter the problem. Out of all practical protocols, the most efficient BFT consensus requires $O(nTL + n^2T)$ communication complexity, where $n$ is number node, $T$ is number of transactions and $L$ is average transaction size. In this work, we propose a new system of BFT fair ordering protocols, Travelers, that substantially reduce the communication complexity. The proposed system of protocols satisfy a new notion of fair ordering, called probabilistic fair ordering, which is an extension to some existing notions of fairness. The new notion allows a small probability of error $\epsilon$, that adversary can insert some transactions at any location in a block, but for the remaining $1-\epsilon$ the a modified version of ordering linearizability holds. Our mechanism neither require a dissemination network nor direct submissions to all consensus nodes. The key innovation comes from a routing protocol, that is both flexible and efficient. We construct a protocol with $O(c\log({n})TL + n^2)$ communication complexity with $\epsilon = 1/n^c$ for some system parameter $c\ge 1$.


Title: Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems
Abstract: Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs.


Title: From Bi-immunity to Absolute Undecidability
Abstract: An infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh-Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.


Title: Matroid Products in Tropical Geometry
Abstract: Symmetric powers of matroids were first introduced by Lovasz and Mason in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor closed and has infinitely many forbidden minors.


Title: Wavenumber-explicit analysis for the Helmholtz $h$-BEM: error estimates and iteration counts for the Dirichlet problem
Abstract: We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber $k$ to have the error in the iterative solution bounded independently of $k$ as $k\rightarrow \infty$ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions as $k \rightarrow \infty$ when the obstacle is nontrapping.


Title: Lower Bounds on Cardinality of Reducts for Decision Tables from Closed Classes
Abstract: In this paper, we consider classes of decision tables closed under removal of attributes (columns) and changing of decisions attached to rows. For decision tables from closed classes, we study lower bounds on the minimum cardinality of reducts, which are minimal sets of attributes that allow us to recognize, for a given row, the decision attached to it. We assume that the number of rows in decision tables from the closed class is not bounded from above by a constant. We divide the set of such closed classes into two families. In one family, only standard lower bounds $\Omega (\log $ ${\rm cl}(T))$ on the minimum cardinality of reducts for decision tables hold, where ${\rm cl}(T)$ is the number of decision classes in the table $T$. In another family, these bounds can be essentially tightened up to $\Omega ({\rm cl}(T)^{1/q})$ for some natural $q$.


Title: Quotients of the braid group that are extensions of the symmetric group
Abstract: We consider normal subgroups $N$ of the braid group $B_n$ such that the quotient $B_n/N$ is an extension of the symmetric group by an abelian group. We show that, if $n\geq 4$, then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions $(n-1,1)$ and $(n-2,2)$, working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.


Title: Nonterminal complexity of some families of infinite regular languages
Abstract: Nonterminal complexity of a context-free language is the smallest possible number of nonterminals in its generating grammar. While in general case nonterminal complexity computation problem is unsolvable, it can be computed for different families of regular languages. In this paper we study nonterminal complexity of some families of infinite regular languages.


Title: The quantitative behaviour of polynomial orbits on nilmanifolds
Abstract: A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".


Title: Proven Distributed Memory Parallelization of Particle Methods
Abstract: We provide a mathematically proven parallelization scheme for particle methods on distributed-memory computer systems. Particle methods are a versatile and widely used class of algorithms for computer simulations and numerical predictions in various applications, ranging from continuum fluid dynamics and granular flows, using methods such as Smoothed Particle Hydrodynamics (SPH) and Discrete Element Methods (DEM) to Molecular Dynamics (MD) simulations in molecular modeling. Particle methods naturally lend themselves to implementation on parallel-computing hardware. So far, however, a mathematical proof of correctness and equivalence to sequential implementations was only available for shared-memory parallelism. Here, we leverage a formal definition of the algorithmic class of particle methods to provide a proven parallelization scheme for distributed-memory computers. We prove that these parallelized particle methods on distributed memory computers are formally equivalent to their sequential counterpart for a well-defined class of particle methods. Notably, the here analyzed parallelization scheme is well-known and commonly used. Our analysis is, therefore, of immediate practical relevance to existing and new parallel software implementations of particle methods and places them on solid theoretical grounds.


Title: Orthogonal matroids over tracts
Abstract: We generalize Baker-Bowler's theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets, and orthogonal vector sets, and establish basic properties on functoriality, duality, and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular if and only if it is representable over $\mathbb{F}_2$ and $\mathbb{F}_3$, which was originally shown by Geelen, and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over $\mathbb{F}_3$ and $\mathbb{F}_4$.


Title: Strong universality, recurrence, and analytic P-ideals in dynamical systems
Abstract: Given a dynamical system $(X,T)$ and a family $\mathsf{I}\subseteq \mathcal{P}(\omega)$ of "small" sets of nonnegative integers, a point $x \in X$ is said to be $\mathsf{I}$-strong universal if for each $y \in X$ there exists a subsequence $(T^nx: n \in A)$ of its orbit which is convergent to $y$ and, in addition, the set of indexes $A$ is "not small," that is, $A\notin \mathsf{I}$. An analoguous definition is given for $\mathsf{I}$-strong recurrence. In this work, we provide several structural properties and relationships between $\mathsf{I}$-strong universality, $\mathsf{I}$-strong recurrence, and the corresponding ordinary notions of $\mathsf{I}$-universality and $\mathsf{I}$-recurrence. As applications, we provide sufficient conditions which ensure the equivalence between the above notions and the property that each nonempty open set contains some cluster point of some orbit. In addition, we show that if $T$ is a homomorphism on a Fr\'{e}chet space $X$ and there exists a dense set of vectors with null orbit, then for each $y \in X$ the set of all vectors $x \in X$ such that $\lim_{n \in A}T^nx=y$ for some $A\subseteq \omega$ with nonzero upper asymptotic density is either empty or comeager. In the special case of linear dynamical systems on Banach spaces with a dense set of uniformly recurrent vectors, we obtain that $T$ is upper frequently hypercyclic if and only if there exists a hypercyclic vector $x \in X$ for which $\lim_{n \in A}T^nx=0$ for some $A\subseteq \omega$ with nonzero upper asymptotic density.


Title: Deep nets for local manifold learning
Abstract: The problem of extending a function $f$ defined on a training data $\mathcal{C}$ on an unknown manifold $\mathbb{X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For $\mathbb{X}$ embedded in a high dimensional ambient Euclidean space $\mathbb{R}^D$, a deep learning algorithm is developed for finding a local coordinate system for the manifold {\bf without eigen--decomposition}, which reduces the problem to the classical problem of function approximation on a low dimensional cube. Deep nets (or multilayered neural networks) are proposed to accomplish this approximation scheme by using the training data. Our methods do not involve such optimization techniques as back--propagation, while assuring optimal (a priori) error bounds on the output in terms of the number of derivatives of the target function. In addition, these methods are universal, in that they do not require a prior knowledge of the smoothness of the target function, but adjust the accuracy of approximation locally and automatically, depending only upon the local smoothness of the target function. Our ideas are easily extended to solve both the pre--image problem and the out--of--sample extension problem, with a priori bounds on the growth of the function thus extended.


Title: Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations
Abstract: We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available.


Title: Distributed Compute-and-Forward Based Relaying Strategies in Multi-User Multi-Relay Networks
Abstract: In this paper, we propose different practical distributed schemes to solve the rank failure problem in the compute and forward (CMF)-based multi-user multi-relay networks without central coordinator, in which the relays have no prior information about each other. First, a new relaying strategy based on CMF, named incremental compute-and-forward (ICMF), is proposed that performs quite well in terms of the outage probability. We show that the distributed ICMF scheme can even outperform the achievable rate of centralized optimal CMF in strong enough inter relay links, with much less complexity. Then, as the second scheme, amplify-forward and compute (AFC) is introduced in which the equations are recovered in the destination rather than in the relays. Finally, ICMF and AFC schemes are combined to present hybrid compute-amplify and forward (HCAF) relaying scheme, which takes advantages of both ICMF, and AFC and improves the performance of the ICMF considerably. We evaluate the performance of the proposed strategies in terms of the outage probability and compare the results with those of the conventional CMF strategy, the Decode and Forward (DF) strategy, and also the centralized optimal CMF. The results indicate the substantial superiority of the proposed schemes compared with the conventional schemes, specially for high number of users and relays.


Title: A proximal point algorithm for sequential feature extraction applications
Abstract: We propose a proximal point algorithm to solve LAROS problem, that is the problem of finding a "large approximately rank-one submatrix". This LAROS problem is used to sequentially extract features in data. We also develop a new stopping criterion for the proximal point algorithm, which is based on the duality conditions of \eps-optimal solutions of the LAROS problem, with a theoretical guarantee. We test our algorithm with two image databases and show that we can use the LAROS problem to extract appropriate common features from these images.


Title: Classifying Words with 3-sort Automata
Abstract: Grammatical inference consists in learning a language or a grammar from data. In this paper, we consider a number of models for inferring a non-deterministic finite automaton (NFA) with 3 sorts of states, that must accept some words, and reject some other words from a given sample. We then propose a transformation from this 3-sort NFA into weighted-frequency and probabilistic NFA, and we apply the latter to a classification task. The experimental evaluation of our approach shows that the probabilistic NFAs can be successfully applied for classification tasks on both real-life and superficial benchmark data sets.


Title: Signal reconstruction from the magnitude of subspace components
Abstract: We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of $p$-fusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program.


Title: Continuum dynamics of the intention field under weakly cohesive social interactions
Abstract: We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. Firstly, we derive a Fokker-Planck type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Secondly, we study conditions under which the Fokker-Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.


Title: On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras
Abstract: Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G_1 and B_1. Furthermore, denote the algebras of polynomial functions on G and g by k[G] and k[g], and similar for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. We show that, under certain mild assumptions, the cohomology groups H^1(G_1,k[g]), H^1(B_1,k[b]), H^1(G_1,k[G]) and H^1(B_1,k[B]) are zero. We also extend all our results to the cohomology for the higher Frobenius kernels.


Title: A Comprehensive Survey on Graph Summarization with Graph Neural Networks
Abstract: As large-scale graphs become more widespread, more and more computational challenges with extracting, processing, and interpreting large graph data are being exposed. It is therefore natural to search for ways to summarize these expansive graphs while preserving their key characteristics. In the past, most graph summarization techniques sought to capture the most important part of a graph statistically. However, today, the high dimensionality and complexity of modern graph data are making deep learning techniques more popular. Hence, this paper presents a comprehensive survey of progress in deep learning summarization techniques that rely on graph neural networks (GNNs). Our investigation includes a review of the current state-of-the-art approaches, including recurrent GNNs, convolutional GNNs, graph autoencoders, and graph attention networks. A new burgeoning line of research is also discussed where graph reinforcement learning is being used to evaluate and improve the quality of graph summaries. Additionally, the survey provides details of benchmark datasets, evaluation metrics, and open-source tools that are often employed in experimentation settings, along with a detailed comparison, discussion, and takeaways for the research community focused on graph summarization. Finally, the survey concludes with a number of open research challenges to motivate further study in this area.


Title: Secure Multiparty Computation with Partial Fairness
Abstract: A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation where parties give their inputs to a trusted party which returns the output of the functionality to all parties. In particular, in the ideal model such computation is fair -- all parties get the output. Cleve (STOC 1986) proved that, in general, fairness is not possible without an honest majority. To overcome this impossibility, Gordon and Katz (Eurocrypt 2010) suggested a relaxed definition -- 1/p-secure computation -- which guarantees partial fairness. For two parties, they construct 1/p-secure protocols for functionalities for which the size of either their domain or their range is polynomial (in the security parameter). Gordon and Katz ask whether their results can be extended to multiparty protocols. We study 1/p-secure protocols in the multiparty setting for general functionalities. Our main result is constructions of 1/p-secure protocols when the number of parties is constant provided that less than 2/3 of the parties are corrupt. Our protocols require that either (1) the functionality is deterministic and the size of the domain is polynomial (in the security parameter), or (2) the functionality can be randomized and the size of the range is polynomial. If the size of the domain is constant and the functionality is deterministic, then our protocol is efficient even when the number of parties is O(log log n) (where n is the security parameter). On the negative side, we show that when the number of parties is super-constant, 1/p-secure protocols are not possible when the size of the domain is polynomial.


Title: Can AI Be as Creative as Humans?
Abstract: Creativity serves as a cornerstone for societal progress and innovation, but its assessment remains a complex and often subjective endeavor. With the rise of advanced generative AI models capable of tasks once reserved for human creativity, the study of AI's creative potential becomes imperative for its responsible development and application. This paper addresses the complexities in defining and evaluating creativity by introducing a new concept called Relative Creativity. Instead of trying to define creativity universally, we shift the focus to whether AI can match the creative abilities of a hypothetical human. This perspective draws inspiration from the Turing Test, expanding upon it to address the challenges and subjectivities inherent in evaluating creativity. This methodological shift facilitates a statistically quantifiable evaluation of AI's creativity, which we term Statistical Creativity. This approach allows for direct comparisons of AI's creative abilities with those of specific human groups. Building on this foundation, we discuss the application of statistical creativity in contemporary prompt-conditioned autoregressive models. In addition to defining and analyzing a measure of creativity, we introduce an actionable training guideline, effectively bridging the gap between theoretical quantification of creativity and practical model training. Through these multifaceted contributions, the paper establishes a cohesive, continuously evolving, and transformative framework for assessing and fostering statistical creativity in AI models.


Title: Super-extensions of tensor algebras and their applications
Abstract: Following arXiv:0909.5586 and arXiv:1411.4125, we construct two super-extensions of the usual tensor algebra through the super-actions of symmetric groups and Hecke algebras respectively. For each extension, we consider a special type of derivations coming from covectors, and study the the space generated, in some special manner, by these derivations and operators from left multiplication by vectors and permutations. Duality theorems of these spaces and the super-actions are proved. As an application, we provide a new proof of the Schur-Sergeev duality theorem, as well as its quantum version.


Title: Closed and asymptotic formulae for harmonic and quadratic harmonic sums
Abstract: We present here a large collection of harmonic and quadratic harmonic sums, that can be useful in applied questions, e.g., probabilistic ones. We find closed-form formulae, that we were not able to locate in the literature.


Title: A taxonomy of estimator consistency on discrete estimation problems
Abstract: We describe a four-level hierarchy mapping both all discrete estimation problems and all estimators on these problems, such that the hierarchy describes each estimator's consistency guarantees on each problem class. We show that no estimator is consistent for all estimation problems, but that some estimators, such as Maximum A Posteriori, are consistent for the widest possible class of discrete estimation problems. For Maximum Likelihood and Approximate Maximum Likelihood estimators we show that they do not provide consistency on as wide a class, but define a sub-class of problems characterised by their consistency. Lastly, we show that some popular estimators, specifically Strict Minimum Message Length, do not provide consistency guarantees even within the sub-class.


Title: Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs
Abstract: We formulate a precise connection between the new Drinfeld presentation of a quantum affine algebra $U_q\widehat{\mathfrak{g}}$ and the new Drinfeld presentation of affine coideal subalgebras of split type recently discovered by Lu and Wang. In particular, we establish a ``factorization formula'', expressing the commuting ``affine Cartan''-type operators $\Theta_{i,k}$ in a coideal subalgebra in terms of the corresponding Drinfeld generators of $U_q\widehat{\mathfrak{g}}$, modulo the ``Drinfeld positive half" of $U_q\widehat{\mathfrak{g}}$. We study the spectra of these operators on finite dimensional representations, and describe them in terms of rational functions with an extra symmetry. These results can be seen as the starting point of a $q$-character theory for affine Kac--Moody quantum symmetric pairs. Additionally, we prove a compatibility result linking Lusztig's and the Lu-Ruan-Wang-Zhang braid group actions.


Title: Twisted Yang-Baxter sets, cohomology theory, and application to knots
Abstract: In this article, we introduce a notion of twisted set-theoretic Yang-Baxter solution, which is a triplet $(X,f,R)$, where $(X,R)$ is a Yang-Baxter set and $f:X \to X$ is an automorphism of $(X,R)$. We present a cohomology theory for it, and use cocycles of twisted biquandles in amalgamation with Alexander numbering to construct state-sum invariant of knots and knotted surfaces. Additionally, we introduce a twisted version of cohomology theory for Yang-Baxter sets and give applications to knot theory.


Title: In Quest of Significance: Identifying Types of Twitter Sentiment Events that Predict Spikes in Sales
Abstract: We study the power of Twitter events to predict consumer sales events by analysing sales for 75 companies from the retail sector and over 150 million tweets mentioning those companies along with their sentiment. We suggest an approach for events identification on Twitter extending existing methodologies of event study. We also propose a robust method for clustering Twitter events into different types based on their shape, which captures the varying dynamics of information propagation through the social network. We provide empirical evidence that through events differentiation based on their shape we can clearly identify types of Twitter events that have a more significant power to predict spikes in sales than the aggregated Twitter signal.


Title: Strategic Bidding Wars in On-chain Auctions
Abstract: The Ethereum block-building process has changed significantly since the emergence of Proposer-Builder Separation. Validators access blocks through a marketplace, where block builders bid for the right to construct the block and earn MEV (Maximal Extractable Value) rewards in an on-chain competition, known as the MEV-boost auction. While more than 90% of blocks are currently built via MEV-Boost, trade-offs between builders' strategic behaviors and auction design remain poorly understood. In this paper we address this gap. We introduce a game-theoretic model for MEV-Boost auctions and use simulations to study different builders' bidding strategies observed in practice. We study various strategic interactions and auction setups and evaluate how the interplay between critical elements such as access to MEV opportunities and improved connectivity to relays impact bidding performance. Our results demonstrate the importance of latency on the effectiveness of builders' strategies and the overall auction outcome from the proposer's perspective.


Title: Introduction of Probabilistic Algebraic Variety
Abstract: Historically, probability theory has been studied for a long time, and Kolmogorov, Levy Ito Kiyoshi, and others have mathematically developed modern probability in conjunction with measurement theory. On the other hand, commutative algebra and algebraic geometry have historically been the subject of interdisciplinary research led by Grothendiek. Many Japanese, notably Matsumura, Hironaka, and Kodaira, have contributed to this field. This paper is an attempt to focus on the research theme of Professor Sumio Watanabe of Tokyo Institute of Technology, "Algebraic Geometry and Probability Theory," from my own perspective. The mathematical theory development starts from Kolmogorov's axioms, and the proof and introduction of "Probabilistic Algebraic Variety" are given. Problems in computation and applications, analysis by computational homology, and unsolved problems in regression problems will be introduced as applications to statistics.


Title: Twisted restricted conformal blocks of vertex operator algebras I: $g$-twisted correlation functions and fusion rules
Abstract: In this paper, we introduce a notion of $g$-twisted restricted conformal block on the three-pointed twisted projective line $\mathfrak{x}\colon\overline{C}\to\mathbb{P^1}$ associated with an untwisted module $M^1$ and the bottom levels of two $g$-twisted modules $M^2$ and $M^3$ over a vertex operator algebra $V$. We show that the space of twisted restricted conformal blocks is isomorphic to the space of $g$-twisted (restricted) correlation functions defined by the same datum and to the space of intertwining operators among these twisted modules. As an application, we derive a twisted version of the Fusion Rules Theorem.


Title: Symbolic Dynamics of Magnetic Bumps
Abstract: For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.


Title: Self-adjointness of the Gaffney Laplacian on vector bundles
Abstract: We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.


Title: Natural real algebraic maps of non-positive codimensions with prescribed images whose boundaries consist of non-singular real algebraic hypersurfaces intersecting with transversality
Abstract: We present new real algebraic maps of non-positive codimensions with prescribed images whose boundaries consist of explicit non-singular real algebraic hypersurfaces intersecting with so-called "transversality". We also have the maps with explicit information on important real polynomials. This gives new construction of explicit real algebraic maps of non-positive codimensions with explicit information on images, preimages and important real polynomials. Thanks to some celebrated theory of Nash, smooth closed manifolds are {\it non-singular} real algebraic manifolds and the zero sets of some real polynomials. In considerable cases, we can approximate smooth functions or more generally, maps, by real algebraic ones. It is in general difficult to have explicit examples. We have constructed some previously. We have ones of some new type with explicit information on global structures of the maps.


Title: Kronecker coefficients and noncommutative super Schur functions
Abstract: The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients where one of the partitions is a hook, recovering previous results of the two authors. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux.


Title: Asymptotic approximations of the continuous Hahn polynomials and their zeros
Abstract: Asymptotic approximations for the continuous Hahn polynomials and their zeros as the degree grows to infinity are established via their three-term recurrence relation. The methods are based on the uniform asymptotic expansions for difference equations developed by Wang and Wong (\textit{Numer. Math.}, 2003) and the matching technique in the complex plane developed by Wang (\textit{J. Approx. Theory}, 2014).


Title: Towards Computing an Optimal Abstraction for Structural Causal Models
Abstract: Working with causal models at different levels of abstraction is an important feature of science. Existing work has already considered the problem of expressing formally the relation of abstraction between causal models. In this paper, we focus on the problem of learning abstractions. We start by defining the learning problem formally in terms of the optimization of a standard measure of consistency. We then point out the limitation of this approach, and we suggest extending the objective function with a term accounting for information loss. We suggest a concrete measure of information loss, and we illustrate its contribution to learning new abstractions.


Title: Economics Arena for Large Language Models
Abstract: Large language models (LLMs) have been extensively used as the backbones for general-purpose agents, and some economics literature suggest that LLMs are capable of playing various types of economics games. Following these works, to overcome the limitation of evaluating LLMs using static benchmarks, we propose to explore competitive games as an evaluation for LLMs to incorporate multi-players and dynamicise the environment. By varying the game history revealed to LLMs-based players, we find that most of LLMs are rational in that they play strategies that can increase their payoffs, but not as rational as indicated by Nash Equilibria (NEs). Moreover, when game history are available, certain types of LLMs, such as GPT-4, can converge faster to the NE strategies, which suggests higher rationality level in comparison to other models. In the meantime, certain types of LLMs can win more often when game history are available, and we argue that the winning rate reflects the reasoning ability with respect to the strategies of other players. Throughout all our experiments, we observe that the ability to strictly follow the game rules described by natural languages also vary among the LLMs we tested. In this work, we provide an economics arena for the LLMs research community as a dynamic simulation to test the above-mentioned abilities of LLMs, i.e. rationality, strategic reasoning ability, and instruction-following capability.


Title: Characterization and Prediction of Deep Learning Workloads in Large-Scale GPU Datacenters
Abstract: Modern GPU datacenters are critical for delivering Deep Learning (DL) models and services in both the research community and industry. When operating a datacenter, optimization of resource scheduling and management can bring significant financial benefits. Achieving this goal requires a deep understanding of the job features and user behaviors. We present a comprehensive study about the characteristics of DL jobs and resource management. First, we perform a large-scale analysis of real-world job traces from SenseTime. We uncover some interesting conclusions from the perspectives of clusters, jobs and users, which can facilitate the cluster system designs. Second, we introduce a general-purpose framework, which manages resources based on historical data. As case studies, we design: a Quasi-Shortest-Service-First scheduling service, which can minimize the cluster-wide average job completion time by up to 6.5x; and a Cluster Energy Saving service, which improves overall cluster utilization by up to 13%.


Title: On dual quaternions, dual split quaternions and Cartan-Schouten metrics on perfect Lie groups
Abstract: We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T*G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T*G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to TSU(2), the set of unit dual split quaternions is isomorphic to T*SL(2,R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T*SO(3). The group SE(2,1) of special rigid displacements of the Minkowski 3-space is isomorphic to T*SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2,1) and to T*G, for any simple Lie group G.


Title: Scalable network reconstruction in subquadratic time
Abstract: Network reconstruction consists in determining the unobserved pairwise couplings between $N$ nodes given only observational data on the resulting behavior that is conditioned on those couplings -- typically a time-series or independent samples from a graphical model. A major obstacle to the scalability of algorithms proposed for this problem is a seemingly unavoidable quadratic complexity of $O(N^2)$, corresponding to the requirement of each possible pairwise coupling being contemplated at least once, despite the fact that most networks of interest are sparse, with a number of non-zero couplings that is only $O(N)$. Here we present a general algorithm applicable to a broad range of reconstruction problems that achieves its result in subquadratic time, with a data-dependent complexity loosely upper bounded by $O(N^{3/2}\log N)$, but with a more typical log-linear complexity of $O(N\log^2N)$. Our algorithm relies on a stochastic second neighbor search that produces the best edge candidates with high probability, thus bypassing an exhaustive quadratic search. In practice, our algorithm achieves a performance that is many orders of magnitude faster than the quadratic baseline, allows for easy parallelization, and thus enables the reconstruction of networks with hundreds of thousands and even millions of nodes and edges.


Title: Joint Matrix-Tensor Factorization for Knowledge Base Inference
Abstract: While several matrix factorization (MF) and tensor factorization (TF) models have been proposed for knowledge base (KB) inference, they have rarely been compared across various datasets. Is there a single model that performs well across datasets? If not, what characteristics of a dataset determine the performance of MF and TF models? Is there a joint TF+MF model that performs robustly on all datasets? We perform an extensive evaluation to compare popular KB inference models across popular datasets in the literature. In addition to answering the questions above, we remove a limitation in the standard evaluation protocol for MF models, propose an extension to MF models so that they can better handle out-of-vocabulary (OOV) entity pairs, and develop a novel combination of TF and MF models. We also analyze and explain the results based on models and dataset characteristics. Our best model is robust, and obtains strong results across all datasets.


Title: Physics-informed Generalizable Wireless Channel Modeling with Segmentation and Deep Learning: Fundamentals, Methodologies, and Challenges
Abstract: Channel modeling is fundamental in advancing wireless systems and has thus attracted considerable research focus. Recent trends have seen a growing reliance on data-driven techniques to facilitate the modeling process and yield accurate channel predictions. In this work, we first provide a concise overview of data-driven channel modeling methods, highlighting their limitations. Subsequently, we introduce the concept and advantages of physics-informed neural network (PINN)-based modeling and a summary of recent contributions in this area. Our findings demonstrate that PINN-based approaches in channel modeling exhibit promising attributes such as generalizability, interpretability, and robustness. We offer a comprehensive architecture for PINN methodology, designed to inform and inspire future model development. A case-study of our recent work on precise indoor channel prediction with semantic segmentation and deep learning is presented. The study concludes by addressing the challenges faced and suggesting potential research directions in this field.


Title: On the support of the Kloosterman paths
Abstract: We obtain statistical results on the possible distribution of all partial sums of a Kloosterman sum modulo a prime, by computing explicitly the support of the limiting random Fourier series of our earlier functional limit theorem for Kloosterman paths.


Title: Entropy-based Probing Beam Selection and Beam Prediction via Deep Learning
Abstract: Hierarchical beam search in mmWave communications incurs substantial training overhead, necessitating deep learning-enabled beam predictions to effectively leverage channel priors and mitigate this overhead. In this study, we introduce a comprehensive probabilistic model of power distribution in beamspace, and formulate the joint optimization problem of probing beam selection and probabilistic beam prediction as an entropy minimization problem. Then, we propose a greedy scheme to iteratively and alternately solve this problem, where a transformer-based beam predictor is trained to estimate the conditional power distribution based on the probing beams and user location within each iteration, and the trained predictor selects an unmeasured beam that minimizes the entropy of remaining beams. To further reduce the number of interactions and the computational complexity of the iterative scheme, we propose a two-stage probing beam selection scheme. Firstly, probing beams are selected from a location-specific codebook designed by an entropy-based criterion, and predictions are made with corresponding feedback. Secondly, the optimal beam is identified using additional probing beams with the highest predicted power values. Simulation results demonstrate the superiority of the proposed schemes compared to hierarchical beam search and beam prediction with uniform probing beams.


Title: Approximating Numerical Flux by Fourier Neural Operators for the Hyperbolic Conservation Laws
Abstract: Classical numerical schemes exist for solving PDEs numerically, and recently, neural network-based methods have been developed. However, methodologies using neural networks, such as PINN and neural operators, lack robustness and generalization power. To compensate for such drawbacks, there are many types of research combining classical numerical schemes and machine learning methods by replacing a small portion of the numerical schemes with neural networks. In this work, we focus on hyperbolic conservation laws and replace numerical fluxes in the numerical schemes by neural operator. For this, we construct losses that are motivated by numerical schemes for conservation laws and approximate numerical flux by FNO. Through experiments, we show that our methodology has advantages of both numerical schemes and FNO by comparing with original methods. For instance, we demonstrate our method gains robustness, resolution invariance property, and feasibility of a data-driven method. Our method especially has the ability to predict continuously in time and generalization power on the out-of-distribution samples, which are challenges to be tackled for existing neural operator methods.


Title: Solution of the Kolmogorov equation for TASEP
Abstract: We provide a direct and elementary proof that the formula obtained in [MQR17] for the TASEP transition probabilities for general (one-sided) initial data solves the Kolmogorov backward equation. The same method yields the solution for the related PushASEP particle system.


Title: Nilpotent dessins: Decomposition theorem and classification of the abelian dessins
Abstract: A map is a 2-cell decomposition of an orientable closed surface. A dessin is a bipartite map with a fixed colouring of vertices. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges, and a regular dessin is symmetric if it admits an additional external symmetry transposing the vertex colours. Regular dessins with nilpotent automorphism groups are investigated. We show that each such dessin is a parallel product of regular dessins whose automorphism groups are the Sylow subgroups. Regular and symmetric dessins with abelian automorphism groups are classified and enumerated.


Title: DGDNN: Decoupled Graph Diffusion Neural Network for Stock Movement Prediction
Abstract: Forecasting future stock trends remains challenging for academia and industry due to stochastic inter-stock dynamics and hierarchical intra-stock dynamics influencing stock prices. In recent years, graph neural networks have achieved remarkable performance in this problem by formulating multiple stocks as graph-structured data. However, most of these approaches rely on artificially defined factors to construct static stock graphs, which fail to capture the intrinsic interdependencies between stocks that rapidly evolve. In addition, these methods often ignore the hierarchical features of the stocks and lose distinctive information within. In this work, we propose a novel graph learning approach implemented without expert knowledge to address these issues. First, our approach automatically constructs dynamic stock graphs by entropy-driven edge generation from a signal processing perspective. Then, we further learn task-optimal dependencies between stocks via a generalized graph diffusion process on constructed stock graphs. Last, a decoupled representation learning scheme is adopted to capture distinctive hierarchical intra-stock features. Experimental results demonstrate substantial improvements over state-of-the-art baselines on real-world datasets. Moreover, the ablation study and sensitivity study further illustrate the effectiveness of the proposed method in modeling the time-evolving inter-stock and intra-stock dynamics.


Title: First mixed Laplace eigenfunctions with no hot spots
Abstract: The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. We then solve this problem for Euclidean triangles and a class of planar domains bounded by the graphs of certain piecewise smooth functions.


Title: Quantum Sets of Compact Quantum Groups
Abstract: Q-system completion can be thought of as a notion of higher idempotent completion of C*-2-categories. We introduce a notion of quantum bi-elements, and study Q-system completion in the context of compact quantum groups. We relate our notion of quantum bi-elements to already known notions of quantum sets and quantum functions, and provide a description of Q-system completion of the C*-2-category of compact quantum groups using quantum bi-elements


Title: Provably Powerful Graph Neural Networks for Directed Multigraphs
Abstract: This paper analyses a set of simple adaptations that transform standard message-passing Graph Neural Networks (GNN) into provably powerful directed multigraph neural networks. The adaptations include multigraph port numbering, ego IDs, and reverse message passing. We prove that the combination of these theoretically enables the detection of any directed subgraph pattern. To validate the effectiveness of our proposed adaptations in practice, we conduct experiments on synthetic subgraph detection tasks, which demonstrate outstanding performance with almost perfect results. Moreover, we apply our proposed adaptations to two financial crime analysis tasks. We observe dramatic improvements in detecting money laundering transactions, improving the minority-class F1 score of a standard message-passing GNN by up to 30%, and closely matching or outperforming tree-based and GNN baselines. Similarly impressive results are observed on a real-world phishing detection dataset, boosting three standard GNNs' F1 scores by around 15% and outperforming all baselines.


Title: Autonomous Reinforcement Learning via Subgoal Curricula
Abstract: Reinforcement learning (RL) promises to enable autonomous acquisition of complex behaviors for diverse agents. However, the success of current reinforcement learning algorithms is predicated on an often under-emphasised requirement -- each trial needs to start from a fixed initial state distribution. Unfortunately, resetting the environment to its initial state after each trial requires substantial amount of human supervision and extensive instrumentation of the environment which defeats the goal of autonomous acquisition of complex behaviors. In this work, we propose Value-accelerated Persistent Reinforcement Learning (VaPRL), which generates a curriculum of initial states such that the agent can bootstrap on the success of easier tasks to efficiently learn harder tasks. The agent also learns to reach the initial states proposed by the curriculum, minimizing the reliance on human interventions into the learning. We observe that VaPRL reduces the interventions required by three orders of magnitude compared to episodic RL while outperforming prior state-of-the art methods for reset-free RL both in terms of sample efficiency and asymptotic performance on a variety of simulated robotics problems.


Title: Stability of strong viscous shock wave under periodic perturbation for 1-D isentropic Navier-Stokes system in the half space
Abstract: In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover, the strength of {the} shock wave could be arbitrarily large. This result essentially improves the previous work " A. Matsumura, M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Ration. Mech. Anal. 146 (1999), no. 1, 1-22." where the strength of shock wave is sufficiently small and the initial periodic oscillations vanish.


Title: Deployment and configuration of MEC apps with Simu5G
Abstract: Multi-access Edge Computing (MEC) is expected to act as the enabler for the integration of 5G (and future 6G) communication technologies with cloud-computing-based capabilities at the edge of the network. This will enable low-latency and context-aware applications for users of such mobile networks. In this paper we describe the implementation of a MEC model for the Simu5G simulator and illustrate how to configure the environment to evaluate MEC applications in both simulation and real-time emulation modes.


Title: An anisotropic partial regularity criterion for the Navier-Stokes equations
Abstract: In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.


Title: From a Kac algebra subfactor to Drinfeld double
Abstract: Given a finite-index and finite-depth subfactor, we define the notion of \textit{quantum double inclusion} - a certain unital inclusion of von Neumann algebras constructed from the given subfactor - which is closely related to that of Ocneanu's asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor $R^H \subset R$ produces Drinfeld double of $H$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H \subset R$ is isomorphic to $R \subset R \rtimes D(H)^{cop}$ for some outer action of $D(H)^{cop}$ on $R$ where $D(H)$ denotes the Drinfeld double of $H$.


Title: The Wave Equation on Lattices and Oscillatory Integrals
Abstract: In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove that the optimal time decay rate of the fundamental solution is of order $|t|^{-\frac{3}{2}}\log |t|$, which is the first extension of P. Schultz's \cite{S98} results in $d=2,3$ to the higher dimension. We also observe that the decay rate for $d=2,3,4$ can be well interpreted by the Newton polyhedra. Moreover, we prove $l^p\rightarrow l^q$ estimates as well as Strichartz estimates and give applications to nonlinear wave equations.


Title: Density estimation using the perceptron
Abstract: We propose a new density estimation algorithm. Given $n$ i.i.d. samples from a distribution belonging to a class of densities on $\mathbb{R}^d$, our estimator outputs any density in the class whose ''perceptron discrepancy'' with the empirical distribution is at most $O(\sqrt{d/n})$. The perceptron discrepancy between two distributions is defined as the largest difference in mass that they place on any halfspace of $\mathbb{R}^d$. It is shown that this estimator achieves expected total variation distance to the truth that is almost minimax optimal over the class of densities with bounded Sobolev norm and Gaussian mixtures. This suggests that regularity of the prior distribution could be an explanation for the efficiency of the ubiquitous step in machine learning that replaces optimization over large function spaces with simpler parametric classes (e.g. in the discriminators of GANs). We generalize the above to show that replacing the ''perceptron discrepancy'' with the generalized energy distance of Sz\'ekeley-Rizzo further improves total variation loss. The generalized energy distance between empirical distributions is easily computable and differentiable, thus making it especially useful for fitting generative models. To the best of our knowledge, it is the first example of a distance with such properties for which there are minimax statistical guarantees.


Title: Human-in-the-Loop Policy Optimization for Preference-Based Multi-Objective Reinforcement Learning
Abstract: Multi-objective reinforcement learning (MORL) aims to find a set of high-performing and diverse policies that address trade-offs between multiple conflicting objectives. However, in practice, decision makers (DMs) often deploy only one or a limited number of trade-off policies. Providing too many diversified trade-off policies to the DM not only significantly increases their workload but also introduces noise in multi-criterion decision-making. With this in mind, we propose a human-in-the-loop policy optimization framework for preference-based MORL that interactively identifies policies of interest. Our method proactively learns the DM's implicit preference information without requiring any a priori knowledge, which is often unavailable in real-world black-box decision scenarios. The learned preference information is used to progressively guide policy optimization towards policies of interest. We evaluate our approach against three conventional MORL algorithms that do not consider preference information and four state-of-the-art preference-based MORL algorithms on two MORL environments for robot control and smart grid management. Experimental results fully demonstrate the effectiveness of our proposed method in comparison to the other peer algorithms.


Title: On the classification of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids
Abstract: In this paper we study a natural generalization of symplectic toric manifolds in the context of regular Poisson manifolds of compact types. To be more precise, we consider a class of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids that we call faithful. Given such a groupoid, we classify its faithful multiplicity-free Hamiltonian actions in terms of what we call Delzant subspaces of its orbit space -- certain `suborbifolds with corners' satisfying the Delzant condition relative to the integral affine orbifold structure of the orbit space. This encompasses both the classification of symplectic toric manifolds (due to Delzant) in terms of Delzant polytopes and the classification of proper Lagrangian fibrations over an integral affine base manifold (due to Duistermaat) in terms of a sheaf cohomology group. Each Delzant subspace comes with an orbifold version of this cohomology, the degree one part of which classifies faithful multiplicity-free Hamiltonian actions with momentum map image equal to the Delzant subspace, provided there exists such an action. The obstruction to existence is encoded by a degree two class in this cohomology: the Lagrangian Dixmier-Douady class. In addition to the above, we introduce another invariant, which leads to a variation of our classification result involving only classical sheaf cohomology and the group cohomology of certain modules for the isotropy groups of the groupoid.


Title: LLM Harmony: Multi-Agent Communication for Problem Solving
Abstract: Large Language Models (LLMs) have revolutionized Natural Language Processing but exhibit limitations, particularly in autonomously addressing novel challenges such as reasoning and problem-solving. Traditional techniques like chain-of-thought prompting necessitate explicit human guidance. This paper introduces a novel multi-agent communication framework, inspired by the CAMEL model, to enhance LLMs' autonomous problem-solving capabilities. The framework employs multiple LLM agents, each with a distinct persona, engaged in role-playing communication, offering a nuanced and adaptable approach to diverse problem scenarios. Extensive experimentation demonstrates the framework's superior performance and adaptability, providing valuable insights into the collaborative potential of multiple agents in overcoming the limitations of individual models.


Title: Concentration-compactness at the mountain pass level in semilinear elliptic problems
Abstract: The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti-Rabinowitz. In these setting, existence of solutions at the mountain pass level is verified under a single penalty condition analogous to that in the Lions' case. Problems on the whole space and problems with critical nonlinearity are considered. Particular attention is given to nonhomogeneous critical nonlinearities that oscillate about the "critical stem".


Title: Local wellposedness for the quasilinear Schrödinger equations via the generalized energy method
Abstract: We study the global Cauchy problem of the quasilinear Schr\"odinger equations, for which KENIG et al. (Invent Math, 2004; Adv Math, 2006) obtained short time local wellposedness with large data by pseudo-differential techniques and viscosity methods, while MARZUOLA et al. (Adv Math, 2012; Kyoto J Math, 2014; Arch Ration Mech Anal, 2021) improved the results by dispersive arguments. In this paper, we introduce the generalized energy method that can close the bounds by combining momentum and energy estimates and derive the results by viscosity methods. The whole arguments basically only involve integration by parts and Sobolev embedding inequalities, just like the classical local existence theorem for semilinear Schr\"odinger equations. For quadratic interaction problem with small data, we derive low regularity local wellposedness in the same function spaces as in the works of Kenig et al. For cubic interaction problem, we obtain the same low regularity results as in Marzuola et al. (Kyoto J Math, 2014).


Title: The bar derived category of a curved dg algebra
Abstract: Curved A-infinity algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A-infinity algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.


Title: Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity
Abstract: In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for minimal gradient graph on exterior domain in dimension 2. Different from the strategies for higher dimensions, instead of the equivalence of Green's function on unbounded domains, we apply a version of iteration methods from Bao--Li--Zhang [Calc.Var PDE, 52(2015), pp. 39-63] that is refined by spherical harmonic expansions to provide a more explicit asymptotic behavior than known results.


Title: Architectural Design for Secure Smart Contract Development
Abstract: As time progresses, the need for more secure applications grows exponentially. The different types of sensitive information that is being transferred virtually has sparked a rise in systems that leverage blockchain. Different sectors are beginning to use this disruptive technology to evaluate the risks and benefits. Sectors like finance, medicine, higher education, and wireless communication have research regarding blockchain. Futhermore, the need for security standards in this area of research is pivotal. In recent past, several attacks on blockchain infrastructures have resulted in hundreds of millions dollars lost and sensitive information compromised. Some of these attacks include DAO attacks, bZx attacks, and Parity Multisignature Wallet Double Attacks which targeted vulnerabilities within smart contracts on the Ethereum network. These attacks exposed the weaknesses of current smart contract development practices which has led to the increase in distrust and adoption of systems that leverage blockchain for its functionality. In this paper, I identify common software vulnerabilities and attacks on blockchain infrastructures, thoroughly detail the smart contract development process and propose a model for ensuring a stronger security standard for future systems leveraging smart contracts. The purpose for proposing a model is to promote trust among end users in the system which is a foundational element for blockchain adoption in the future.


Title: The impact of the limit $q$-Durrmeyer operator on continuous functions
Abstract: The limit $q$-Durrmeyer operator, $D_{\infty,q},$ was introduced and its approximation properties were investigated by V. Gupta in 2008 during a study of $q$-analogues for the Bernstein-Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of $D_{\infty,q}$. The interrelation between the analytic properties of a function $f$ and the rate of growth for $D_{\infty,q}f$ are established, and the sharpness of the obtained results are demonstrated.


Title: Characterizing BigBench queries, Hive, and Spark in multi-cloud environments
Abstract: BigBench is the new standard (TPCx-BB) for benchmarking and testing Big Data systems. The TPCx-BB specification describes several business use cases -- queries -- which require a broad combination of data extraction techniques including SQL, Map/Reduce (M/R), user code (UDF), and Machine Learning to fulfill them. However, currently, there is no widespread knowledge of the different resource requirements and expected performance of each query, as is the case to more established benchmarks. At the same time, cloud providers currently offer convenient on-demand managed big data clusters (PaaS) with a pay-as-you-go model. In PaaS, analytical engines such as Hive and Spark come ready to use, with a general-purpose configuration and upgrade management. The study characterizes both the BigBench queries and the out-of-the-box performance of Spark and Hive versions in the cloud. At the same time, comparing popular PaaS offerings in terms of reliability, data scalability (1GB to 10TB), versions, and settings from Azure HDinsight, Amazon Web Services EMR, and Google Cloud Dataproc. The query characterization highlights the similarities and differences in Hive an Spark frameworks, and which queries are the most resource consuming according to CPU, memory, and I/O. Scalability results show how there is a need for configuration tuning in most cloud providers as data scale grows, especially with Sparks memory usage. These results can help practitioners to quickly test systems by picking a subset of the queries which stresses each of the categories. At the same time, results show how Hive and Spark compare and what performance can be expected of each in PaaS.


Title: Quota Trees
Abstract: We introduce the notion of quota trees in directed graphs. Given a nonnegative integer ``quota'' for each vertex of a directed multigraph $G$, a quota tree is an immersed rooted tree which hits each vertex of $G$ the prescribed number of times. When the quotas are all one, the tree is actually embedded and we recover the usual notion of a spanning arborescence (directed spanning tree). The usual algorithms which produce spanning arborescences with various properties typically have (sometimes more complicated) ``quota'' analogues. Our original motivation for studying quota trees was the problem of characterizing the sizes of the Myhill-Nerode equivalence classes in a connected deterministic finite-state automaton recognizing a given regular language. We show that the obstruction to realizing a given set of M-N class sizes is precisely the existence of a suitable quota tree. In this paper we develop the basic theory of quota trees. We give necessary and sufficient conditions for the existence of a quota tree (or forest) over a given directed graph with specified quotas, solving the M-N class size problem as a special case. We discuss some potential applications of quota trees and forests, and connect them to the $k$ lightest paths problem. We give two proofs of the main theorem: one based on an algorithmic loop invariant, and one based on direct enumeration of quota trees. For the latter, we use Lagrange inversion to derive a formula which vastly generalizes both the matrix-tree theorem and Cayley's formula for counting labeled trees. We give an efficient algorithm to sample uniformly from the set of forests with given quotas, as well as a generalization of Edmonds' algorithm for computing a minimum-weight quota forest.


Title: Standing waves with prescribed $L^2$-norm to nonlinear Schrödinger equations with combined inhomogeneous nonlinearities
Abstract: In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $\mu =\pm 1$, $2<q<p<{2(N-b)}/{(N-2)^+}$, $0<b<\min\{2, N\}$ and the parameter $\lambda \in \R$ appearing as Lagrange multiplier is unknown. In the mass subcritical case, we establish the compactness of any minimizing sequence to the minimization problem given by the underlying energy functional restricted on the constraint. As a consequence of the compactness of any minimizing sequence, orbital stability of minimizers is derived. In the mass critical and supercritical cases, we investigate the existence, radial symmetry and orbital instability of solutions. Meanwhile, we consider the existence, radial symmetry and algebraical decay of ground states to the corresponding zero mass equation with defocusing perturbation. In addition, dynamical behaviors of solutions to the Cauchy problem for the associated dispersive equation are discussed.


Title: Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings
Abstract: This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the $(k,q)$-colourings of a graph $G$ can be obtained by successively recolouring a single vertex provided $k/q\geq 2col(G)$ along the lines of Cereceda, van den Heuvel and Johnson's result for $k$-colourings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colourings coincide. As a corollary, we obtain an Albertson-type extension theorem for $(k,q)$-precolourings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs $G$ for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.


Title: Beyond Regrets: Geometric Metrics for Bayesian Optimization
Abstract: Bayesian optimization is a principled optimization strategy for a black-box objective function. It shows its effectiveness in a wide variety of real-world applications such as scientific discovery and experimental design. In general, the performance of Bayesian optimization is assessed by regret-based metrics such as instantaneous, simple, and cumulative regrets. These metrics only rely on function evaluations, so that they do not consider geometric relationships between query points and global solutions, or query points themselves. Notably, they cannot discriminate if multiple global solutions are successfully found. Moreover, they do not evaluate Bayesian optimization's abilities to exploit and explore a search space given. To tackle these issues, we propose four new geometric metrics, i.e., precision, recall, average degree, and average distance. These metrics allow us to compare Bayesian optimization algorithms considering the geometry of both query points and global optima, or query points. However, they are accompanied by an extra parameter, which needs to be carefully determined. We therefore devise the parameter-free forms of the respective metrics by integrating out the additional parameter. Finally, we empirically validate that our proposed metrics can provide more convincing interpretation and understanding of Bayesian optimization algorithms from distinct perspectives, compared to the conventional metrics.


Title: Singularity-agnostic incomplete U-statistics for testing polynomial constraints in Gaussian covariance matrices
Abstract: Testing the goodness-of-fit of a model with its defining functional constraints in the parameters could date back to Spearman (1927), who analyzed the famous "tetrad" polynomial in the covariance matrix of the observed variables in a single-factor model. Despite its long history, the Wald test typically employed to operationalize this approach could produce very inaccurate test sizes in many situations, even when the regular conditions for the classical normal asymptotics are met and a very large sample is available. Focusing on testing a polynomial constraint in a Gaussian covariance matrix, we obtained a new understanding of this baffling phenomenon: When the null hypothesis is true but "near-singular", the standardized Wald test exhibits slow weak convergence, owing to the sophisticated dependency structure inherent to the underlying U-statistic that ultimately drives its limiting distribution; this can also be rigorously explained by a key ratio of moments encoded in the Berry-Esseen bound quantifying the normal approximation error involved. As an alternative, we advocate the use of an incomplete U-statistic to mildly tone down the dependence thereof and render the speed of convergence agnostic to the singularity status of the hypothesis. In parallel, we develop a Berry-Esseen bound that is mathematically descriptive of the singularity-agnostic nature of our standardized incomplete U-statistic, using some of the finest exponential-type inequalities in the literature.


Title: Compatibility of Hodge Theory on Alexander Modules
Abstract: Let $U$ be a smooth connected complex algebraic variety, and let $f\colon U\to \mathbb C^*$ be an algebraic map. To the pair $(U,f)$ one can associate an infinite cyclic cover $U^f$, and (homology) Alexander modules are defined as the homology groups of this cover. In two recent works, the first of which is joint with Geske, Maxim and Wang, we developed two different ways to put a mixed Hodge structure on Alexander modules. Since they are not finite dimensional in general, each approach replaces the Alexander module by a different finite dimensional module: one of them takes the torsion submodule, the other takes finite dimensional quotients, and the constructions are not directly comparable. In this note, we show that both constructions are compatible, in the sense that the map from the torsion to the quotients is a mixed Hodge structure morphism.


Title: HEMlets PoSh: Learning Part-Centric Heatmap Triplets for 3D Human Pose and Shape Estimation
Abstract: Estimating 3D human pose from a single image is a challenging task. This work attempts to address the uncertainty of lifting the detected 2D joints to the 3D space by introducing an intermediate state-Part-Centric Heatmap Triplets (HEMlets), which shortens the gap between the 2D observation and the 3D interpretation. The HEMlets utilize three joint-heatmaps to represent the relative depth information of the end-joints for each skeletal body part. In our approach, a Convolutional Network (ConvNet) is first trained to predict HEMlets from the input image, followed by a volumetric joint-heatmap regression. We leverage on the integral operation to extract the joint locations from the volumetric heatmaps, guaranteeing end-to-end learning. Despite the simplicity of the network design, the quantitative comparisons show a significant performance improvement over the best-of-grade methods (e.g. $20\%$ on Human3.6M). The proposed method naturally supports training with "in-the-wild" images, where only weakly-annotated relative depth information of skeletal joints is available. This further improves the generalization ability of our model, as validated by qualitative comparisons on outdoor images. Leveraging the strength of the HEMlets pose estimation, we further design and append a shallow yet effective network module to regress the SMPL parameters of the body pose and shape. We term the entire HEMlets-based human pose and shape recovery pipeline HEMlets PoSh. Extensive quantitative and qualitative experiments on the existing human body recovery benchmarks justify the state-of-the-art results obtained with our HEMlets PoSh approach.


Title: Deep Demosaicing for Edge Implementation
Abstract: Most digital cameras use sensors coated with a Color Filter Array (CFA) to capture channel components at every pixel location, resulting in a mosaic image that does not contain pixel values in all channels. Current research on reconstructing these missing channels, also known as demosaicing, introduces many artifacts, such as zipper effect and false color. Many deep learning demosaicing techniques outperform other classical techniques in reducing the impact of artifacts. However, most of these models tend to be over-parametrized. Consequently, edge implementation of the state-of-the-art deep learning-based demosaicing algorithms on low-end edge devices is a major challenge. We provide an exhaustive search of deep neural network architectures and obtain a pareto front of Color Peak Signal to Noise Ratio (CPSNR) as the performance criterion versus the number of parameters as the model complexity that beats the state-of-the-art. Architectures on the pareto front can then be used to choose the best architecture for a variety of resource constraints. Simple architecture search methods such as exhaustive search and grid search require some conditions of the loss function to converge to the optimum. We clarify these conditions in a brief theoretical study.