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Title: Periodicity of Adams operations on the Green ring of a finite group
Abstract: The Adams operations $\psi_\Lambda^n$ and $\psi_S^n$ on the Green ring of a group $G$ over a field $K$ provide a framework for the study of the exterior powers and symmetric powers of $KG$-modules. When $G$ is finite and $K$ has prime characteristic $p$ we show that $\psi_\Lambda^n$ and $\psi_S^n$ are periodic in $n$ if and only if the Sylow $p$-subgroups of $G$ are cyclic. In the case where $G$ is a cyclic $p$-group we find the minimum periods and use recent work of Symonds to express $\psi_S^n$ in terms of $\psi_\Lambda^n$. | math | 1 |
Title: Hybrid and Oriented Harmonic Potentials for Safe Task Execution in Unknown Environment
Abstract: Harmonic potentials provide globally convergent potential fields that are provably free of local minima. Due to its analytical format, it is particularly suitable for generating safe and reliable robot navigation policies. However, for complex environments that consist of a large number of overlapping non-sphere obstacles, the computation of associated transformation functions can be tedious. This becomes more apparent when: (i) the workspace is initially unknown and the underlying potential fields are updated constantly as the robot explores it; (ii) the high-level mission consists of sequential navigation tasks among numerous regions, requiring the robot to switch between different potentials. Thus, this work proposes an efficient and automated scheme to construct harmonic potentials incrementally online as guided by the task automaton. A novel two-layer harmonic tree (HT) structure is introduced that facilitates the hybrid combination of oriented search algorithms for task planning and harmonic-based navigation controllers for non-holonomic robots. Both layers are adapted efficiently and jointly during online execution to reflect the actual feasibility and cost of navigation within the updated workspace. Global safety and convergence are ensured both for the high-level task plan and the low-level robot trajectory. Known issues such as oscillation or long-detours for purely potential-based methods and sharp-turns or high computation complexity for purely search-based methods are prevented. Extensive numerical simulation and hardware experiments are conducted against several strong baselines. | cs | 0 |
Title: Morphisms of rational motivic homotopy types
Abstract: We investigate several interrelated foundational questions pertaining to the study of motivic dga's of Dan-Cohen--Schlank [8] and Iwanari [13]. In particular, we note that morphisms of motivic dga's can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga's is a certain limit of such spectra; we give an explicit formula for this limit --- a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call ``motivic Chow coalgebras''. We discuss the relationship between motivic Chow coalgebras and motivic dga's of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite \'etale over a number field, morphisms of associated motivic dga's are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim's relative unipotent section conjecture, hence as an ounce of evidence for the latter. | math | 1 |
Title: Symplectic leaves in projective spaces of bundle extensions
Abstract: Fix a stable degree-$n$ rank-$k$ bundle $\mathcal{F}$ on a complex elliptic curve for (coprime) $1\le k<n\ge 3$. We identify the symplectic leaves of the Poisson structure introduced independently by Polishchuk and Feigin-Odesskii on $\mathbb{P}^{n-1}\cong \mathbb{P}\mathrm{Ext}^1(\mathcal{F},\mathcal{O})$ as precisely the loci classifying extensions $0\to \mathcal{O}\to \mathcal{E}\to \mathcal{F}\to 0$ with $\mathcal{E}$ fitting into a fixed isomorphism class, verifying a claim of Feigin-Odesskii. We also classify the bundles $\mathcal{E}$ which do fit into such extensions in geometric / combinatorial terms, involving their Harder-Narasimhan polygons introduced by Shatz. | math | 0 |
Title: The Balkans Continued Fraction
Abstract: In a previous escapade we gave a collection of continued fractions involving Catalan's constant. This paper provides more general formulae governing those continued fractions. Having distinguished different cases associated to regions in the plan, we nickname those continued fractions \enquote{The Balkans} as they divide into areas which are related but still different in nature. Because we do not provide formal proofs of those machine-constructed formulae we do not claim them to be theorems. Still, each and every proposed formula was extensively tested numerically. | math | 0 |
Title: Rational Construction of Stochastic Numerical Methods for Molecular Sampling
Abstract: In this article, we focus on the sampling of the configurational Gibbs-Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular $N$-body system modelled at constant temperature. We show how a formal series expansion of the invariant measure of a Langevin dynamics numerical method can be obtained in a straightforward way using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property (4th order accuracy where only 2nd order would be expected) of one method in the high friction limit; this method, moreover, can be reduced to a simple modification of the Euler-Maruyama method for Brownian dynamics involving a non-Markovian (coloured noise) random process. In the Brownian dynamics case, 2nd order accuracy of the invariant density is achieved. All methods considered are efficient for molecular applications (requiring one force evaluation per timestep) and of a simple form. In fully resolved (long run) molecular dynamics simulations, for our favoured method, we observe up to two orders of magnitude improvement in configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared to common alternative methods. | math | 1 |
Title: Stochastic Approach for Price Optimization Problems with Decision-dependent Uncertainty
Abstract: Price determination is a central research topic of revenue management in marketing. The important aspect in pricing is controlling the stochastic behavior of demand, and the previous studies have tackled price optimization problems with uncertainties. However, many of those studies assumed that uncertainties are independent of decision variables (i.e., prices) and did not consider situations where demand uncertainty depends on price. Although some price optimization studies have dealt with decision-dependent uncertainty, they make application-specific assumptions in order to obtain an optimal solution or an approximation solution. To handle a wider range of applications with decision-dependent uncertainty, we propose a general non-convex stochastic optimization formulation. This approach aims to maximize the expectation of a revenue function with respect to a random variable representing demand under a decision-dependent distribution. We derived an unbiased stochastic gradient estimator by using a well-tuned variance reduction parameter and used it for a projected stochastic gradient descent method to find a stationary point of our problem. We conducted synthetic experiments and simulation experiments with real data on a retail service application. The results show that the proposed method outputs solutions with higher total revenues than baselines. | math | 0 |
Title: The rational (non-)formality of the non-3-equal manifolds
Abstract: Let $M^{(k)}_{d}(n)$ be the manifold of $n$-tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non-$k$-equal coordinates. We show that, for $d\geq2$, $M^{(3)}_{d}(n)$ is rationally formal if and only if $n\leq 6$. This stands in sharp contrast with the fact that all classical configuration spaces $M^{(2)}_d(n)=\text{Conf}(\hspace{.2mm}\mathbb{R}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non formality of $M^{(3)}_{d}(n)$ for $n>6$ is established via detection of non-trivial triple Massey products assessed through Poincar\'e duality. | math | 0 |
Title: Inner conjugate pair of Hadamard subfactors and vertex model
Abstract: We show that any pair of Hadamard subfactors arising from complex Hadamard matrices of order 3 are either equal or inner conjugate. If the pair of Hadamard subfactors are distinct, their intersection is shown to be a subfactor of the hyperfinite type $II_1$ factor $R$. We compute its first relative commutant and characterize this subfactor by identifying it with a vertex model subfactor of the Krishnan-Sunder type. A few key invariants, including the Pimsner-Popa probabilistic number, the angle, and the Connes-St{\o}rmer relative entropy for the pair of Hadamard subfactors are computed to understand their relative position. | math | 0 |
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. | math | 0 |
Title: A note on odd partition numbers
Abstract: Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$ assert that $$ p(\ell n+\delta_{\ell})\equiv 0\pmod{\ell}, $$ where $0<\delta_{\ell}<\ell$ satisfies $24\delta_{\ell}\equiv 1\pmod{\ell}.$ By proving Subbarao's Conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes $\ell \geq 5,$ we give a new proof of the conclusion that there are infinitely many $m$ for which $p(\ell m+\delta_{\ell})$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such $m$ satisfies $m<(\ell^2-1)/24,$ representing a significant improvement to the previous bound. | math | 0 |
Title: Algebraic structures in set-theoretic Yang-Baxter & reflection equations
Abstract: We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the solutions of the set theoretic Yang-Baxter equation. We then restrict our attention on involutive solutions and consider lambda parametric set-theoretic solutions of the Yang-Baxter equation and we extract the associated quantum algebra. We also discuss the notion of the Drinfeld twist for involutive solutions and their relation to the Yangian. We next focus on reflections and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the symmetric group. We show that there exists a ``reflection'' finite sub-algebra for some special choice of reflection maps. | math | 0 |
Title: Fano threefolds with noncyclic torsion in the divisor class group
Abstract: In this note we study Fano threefolds with noncyclic torsion in the divisor class group. Since they can all be obtained as quotients of Fano threefolds, we get also all examples that can be obtained as quotients of low codimension Fanos in the weighted projective space. | math | 1 |
Title: Knot concordance, the point class in instanton homology and Donaldson invariants
Abstract: We define an invariant ${\varphi}$ for knots in the 3-sphere by means of Donaldson invariants and Floer's instanton homology. Some basic properties of this invariant are established and it is shown that ${\varphi}$ coincides with a special case of an invariant defined by Froyshov | math | 0 |
Title: A matrix concentration inequality for products
Abstract: We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Z_n = \left(I_d-\alpha X_n\right)\left(I_d-\alpha X_{n-1}\right)\cdots \left(I_d-\alpha X_1\right), \end{equation} where $\left\{X_k \right\}_{k=1}^{+\infty}$ is a sequence of bounded independent random positive semidefinite matrices with common expectation $\mathbb{E}\left[X_k\right]=\Sigma$. Under these assumptions, we show that, for small enough positive $\alpha$, $Z_n$ satisfies the concentration inequality \begin{equation}\label{eq:CTbound} \mathbb{P}\left(\left\Vert Z_n-\mathbb{E}\left[Z_n\right]\right\Vert \geq t\right) \leq 2d^2\cdot\exp\left(\frac{-t^2}{\alpha \sigma^2} \right) \quad \text{for all } t\geq 0, \end{equation} where $\sigma^2$ denotes a variance parameter. | math | 1 |
Title: A limit theory for controlled McKean-Vlasov SPDEs
Abstract: We develop a limit theory for controlled mean field stochastic partial differential equations in a variational framework. More precisely, we prove existence results for mean field limits and particle approximations, and we establish a set-valued propagation of chaos result which shows that sets of empirical distributions converge to sets of mean field limits in the Hausdorff metric topology. Further, we discuss limit theorems related to stochastic optimal control theory. To illustrate our findings, we apply them to a controlled interacting particle system of stochastic porous media equations. | math | 0 |
Title: Von Neumann entropy of the angle operator between a pair of intermediate subalgebras
Abstract: Given a pair of intermediate $C^*$-subalgebras of a unital inclusion of simple $C^*$-algebras with a conditional expectation of finite Watatani index, we discuss the corresponding angle operator and its Fourier transform. We provide a calculable formula for the von Neumann entropy of the (Fourier) dual angle operator for a large class of quadruple of simple $C^*$-algebras. | math | 0 |
Title: Quantum cohomology of minuscule homogeneous spaces
Abstract: We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov-Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincar\'{e} duality. In particular we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties. | math | 1 |
Title: An Open and Comprehensive Pipeline for Unified Object Grounding and Detection
Abstract: Grounding-DINO is a state-of-the-art open-set detection model that tackles multiple vision tasks including Open-Vocabulary Detection (OVD), Phrase Grounding (PG), and Referring Expression Comprehension (REC). Its effectiveness has led to its widespread adoption as a mainstream architecture for various downstream applications. However, despite its significance, the original Grounding-DINO model lacks comprehensive public technical details due to the unavailability of its training code. To bridge this gap, we present MM-Grounding-DINO, an open-source, comprehensive, and user-friendly baseline, which is built with the MMDetection toolbox. It adopts abundant vision datasets for pre-training and various detection and grounding datasets for fine-tuning. We give a comprehensive analysis of each reported result and detailed settings for reproduction. The extensive experiments on the benchmarks mentioned demonstrate that our MM-Grounding-DINO-Tiny outperforms the Grounding-DINO-Tiny baseline. We release all our models to the research community. Codes and trained models are released at https://github.com/open-mmlab/mmdetection/configs/mm_grounding_dino. | cs | 0 |
Title: Policy-regularized Offline Multi-objective Reinforcement Learning
Abstract: In this paper, we aim to utilize only offline trajectory data to train a policy for multi-objective RL. We extend the offline policy-regularized method, a widely-adopted approach for single-objective offline RL problems, into the multi-objective setting in order to achieve the above goal. However, such methods face a new challenge in offline MORL settings, namely the preference-inconsistent demonstration problem. We propose two solutions to this problem: 1) filtering out preference-inconsistent demonstrations via approximating behavior preferences, and 2) adopting regularization techniques with high policy expressiveness. Moreover, we integrate the preference-conditioned scalarized update method into policy-regularized offline RL, in order to simultaneously learn a set of policies using a single policy network, thus reducing the computational cost induced by the training of a large number of individual policies for various preferences. Finally, we introduce Regularization Weight Adaptation to dynamically determine appropriate regularization weights for arbitrary target preferences during deployment. Empirical results on various multi-objective datasets demonstrate the capability of our approach in solving offline MORL problems. | cs | 0 |
Title: Spiker+: a framework for the generation of efficient Spiking Neural Networks FPGA accelerators for inference at the edge
Abstract: Including Artificial Neural Networks in embedded systems at the edge allows applications to exploit Artificial Intelligence capabilities directly within devices operating at the network periphery. This paper introduces Spiker+, a comprehensive framework for generating efficient, low-power, and low-area customized Spiking Neural Networks (SNN) accelerators on FPGA for inference at the edge. Spiker+ presents a configurable multi-layer hardware SNN, a library of highly efficient neuron architectures, and a design framework, enabling the development of complex neural network accelerators with few lines of Python code. Spiker+ is tested on two benchmark datasets, the MNIST and the Spiking Heidelberg Digits (SHD). On the MNIST, it demonstrates competitive performance compared to state-of-the-art SNN accelerators. It outperforms them in terms of resource allocation, with a requirement of 7,612 logic cells and 18 Block RAMs (BRAMs), which makes it fit in very small FPGA, and power consumption, draining only 180mW for a complete inference on an input image. The latency is comparable to the ones observed in the state-of-the-art, with 780us/img. To the authors' knowledge, Spiker+ is the first SNN accelerator tested on the SHD. In this case, the accelerator requires 18,268 logic cells and 51 BRAM, with an overall power consumption of 430mW and a latency of 54 us for a complete inference on input data. This underscores the significance of Spiker+ in the hardware-accelerated SNN landscape, making it an excellent solution to deploy configurable and tunable SNN architectures in resource and power-constrained edge applications. | cs | 0 |
Title: The infinite Arnoldi exponential integrator for linear inhomogeneous ODEs
Abstract: Exponential integrators that use Krylov approximations of matrix functions have turned out to be efficient for the time-integration of certain ordinary differential equations (ODEs). This holds in particular for linear homogeneous ODEs, where the exponential integrator is equivalent to approximating the product of the matrix exponential and a vector. In this paper, we consider linear inhomogeneous ODEs, $y'(t)=Ay(t)+g(t)$, where the function $g(t)$ is assumed to satisfy certain regularity conditions. We derive an algorithm for this problem which is equivalent to approximating the product of the matrix exponential and a vector using Arnoldi's method. The construction is based on expressing the function $g(t)$ as a linear combination of given basis functions $[\phi_i]_{i=0}^\infty$ with particular properties. The properties are such that the inhomogeneous ODE can be restated as an infinite-dimensional linear homogeneous ODE. Moreover, the linear homogeneous infinite-dimensional ODE has properties that directly allow us to extend a Krylov method for finite-dimensional linear ODEs. Although the construction is based on an infinite-dimensional operator, the algorithm can be carried out with operations involving matrices and vectors of finite size. This type of construction resembles in many ways the infinite Arnoldi method for nonlinear eigenvalue problems. We prove convergence of the algorithm under certain natural conditions, and illustrate properties of the algorithm with examples stemming from the discretization of partial differential equations. | math | 1 |
Title: A Unified Framework for Rank-based Loss Minimization
Abstract: The empirical loss, commonly referred to as the average loss, is extensively utilized for training machine learning models. However, in order to address the diverse performance requirements of machine learning models, the use of the rank-based loss is prevalent, replacing the empirical loss in many cases. The rank-based loss comprises a weighted sum of sorted individual losses, encompassing both convex losses like the spectral risk, which includes the empirical risk and conditional value-at-risk, and nonconvex losses such as the human-aligned risk and the sum of the ranked range loss. In this paper, we introduce a unified framework for the optimization of the rank-based loss through the utilization of a proximal alternating direction method of multipliers. We demonstrate the convergence and convergence rate of the proposed algorithm under mild conditions. Experiments conducted on synthetic and real datasets illustrate the effectiveness and efficiency of the proposed algorithm. | math | 0 |
Title: Cost Minimization in Multi-cloud Systems with Runtime Microservice Re-orchestration
Abstract: Multi-cloud systems facilitate a cost-efficient and geographically-distributed deployment of microservice-based applications by temporary leasing virtual nodes with diverse pricing models. To preserve the cost-efficiency of multi-cloud deployments, it is essential to redeploy microservices onto the available nodes according to a dynamic resource configuration, which is often performed to better accommodate workload variations. However, this approach leads to frequent service disruption since applications are continuously shutdown and redeployed in order to apply the new resource assignment. To overcome this issue, we propose a re-orchestration scheme that migrates microservice at runtime based on a rolling update scheduling logic. Specifically, we propose an integer linear optimization problem that minimizes the cost associated to multi-cloud virtual nodes and that ensures that delay-sensitive microservices are co-located on the same regional cluster. The resulting rescheduling order guarantees no service disruption by repacking microservices between the available nodes without the need to turn off the outdated microservice instance before redeploying the updated version. In addition, we propose a two-step heuristic scheme that effectively approximates the optimal solution at the expense of close-to-zero service disruption and QoS violation probability. Results show that proposed schemes achieve better performance in terms of cost mitigation, low service disruption and low QoS violation probability compared to baseline schemes replicating Kubernetes scheduler functionalities. | cs | 0 |
Title: Log-concave Density Estimation with Independent Components
Abstract: We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(\epsilon^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $\epsilon$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use. | math | 0 |
Title: Subdifferentials of convex matrix-valued functions
Abstract: Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner partial order can have a complicated structure and might be very difficult to compute even in simple cases. The aim of this paper is to study subdifferential calculus for such functions and properties of their subdifferentials. We show that many standard results from convex analysis no longer hold true in the matrix-valued case. For example, in this case the subdifferential of the sum is not equal to the sum of subdifferentials, the Clarke subdifferential is not equal to the subdifferential in the sense of convex analysis, etc. Nonetheless, it is possible to provide simple rules for computing nonempty subsets of subdifferentials (in particular, individual subgradients) of convex matrix-valued functions in the general case and to completely describe subdifferentials of such functions defined on the real line. As a by-product of our analysis, we derive some interesting properties of convex matrix-valued functions, e.g. we show that if such function is nonsmooth, then its diagonal elements must be nonsmooth as well. | math | 0 |
Title: Nodal solutions for Neumann systems with gradient dependence
Abstract: We consider the following convective Neumann systems:\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-\Delta_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+\delta_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) & \text{in}\;\Omega,\\ -\Delta _{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+\delta_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2)&\text{in}\;\Omega, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial \eta }=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial \eta}&\text{on}\;\partial\Omega,\end{array}\right.\end{equation*}where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$) with a smooth boundary $\partial\Omega$,$\delta_1,\,\delta_2 >0$ are small parameters, $\eta$ is the outward unit vector normal to $\partial \Omega,$ $f_1,\,f_2:\Omega\times\mathbb{R}^2\times\mathbb{R}^{2N}\rightarrow \mathbb{R}$ are Carath\'{e}odory functions that satisfy certain growth conditions, and $\Delta _{p_i}$ ($1<p_i<N,$ for $i=1,2$) are the $p$-Laplace operators $\Delta _{p_i}u_i=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$,for every $\,u_i\in W^{1,p_i}(\Omega).$ In order to prove the existence of solutions to such systems, we use a sub-supersolution method. We also obtain nodal solutions by constructing appropriate sub-solution and super-solution pairs. To the best of our knowledge, such systems have not been studied yet. | math | 0 |
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. | cs | 1 |
Title: Cubic bent functions outside the completed Maiorana-McFarland class
Abstract: In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in $n$ variables for all $n\ge 10$. Moreover, we show that for almost all values of $n$, these functions can simultaneously be homogeneous and have no affine derivatives. | math | 1 |
Title: Universal Homotopy Theories and Associated Homological Algebras
Abstract: Let $\mathscr{C}$ be a small category. For every commutative ring $R$ with unity, we associate an $R\mathrm{-linear}$ abelian category with the universal homotopy category of $\mathscr{C}$, where we can do the corresponding homological algebra. | math | 0 |
Title: The Drinfel'd Double and Twisting in Stringy Orbifold Theory
Abstract: This paper exposes the fundamental role that the Drinfel'd double $\dkg$ of the group ring of a finite group $G$ and its twists $\dbkg$, $\beta \in Z^3(G,\uk)$ as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that $G$--Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of $\dkg$--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold $K$--theory of global quotient given by the inertia variety of a point with a $G$ action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full $K$--theory of the stack $[pt/G]$. Finally, we show how one can use the co-cycles $\beta$ above to twist a) the global orbifold $K$--theory of the inertia of a global quotient and more importantly b) the stacky $K$--theory of a global quotient $[X/G]$. This corresponds to twistings with a special type of 2--gerbe. | math | 1 |
Title: Bring Metric Functions into Diffusion Models
Abstract: We introduce a Cascaded Diffusion Model (Cas-DM) that improves a Denoising Diffusion Probabilistic Model (DDPM) by effectively incorporating additional metric functions in training. Metric functions such as the LPIPS loss have been proven highly effective in consistency models derived from the score matching. However, for the diffusion counterparts, the methodology and efficacy of adding extra metric functions remain unclear. One major challenge is the mismatch between the noise predicted by a DDPM at each step and the desired clean image that the metric function works well on. To address this problem, we propose Cas-DM, a network architecture that cascades two network modules to effectively apply metric functions to the diffusion model training. The first module, similar to a standard DDPM, learns to predict the added noise and is unaffected by the metric function. The second cascaded module learns to predict the clean image, thereby facilitating the metric function computation. Experiment results show that the proposed diffusion model backbone enables the effective use of the LPIPS loss, leading to state-of-the-art image quality (FID, sFID, IS) on various established benchmarks. | cs | 0 |
Title: HCIZ integral formula as unitarity of a canonical map between reproducing kernel spaces
Abstract: In this article we prove that the Harish-Chandra-Itzykson-Zuber (HCIZ) integral formula is equivalent to the unitarity of a canonical map between invariant subspaces of Segal-Bargmann spaces. As a consequence, we provide alternative proofs of the HCIZ integral and other results. | math | 0 |
Title: Mod-$p$ isogeny classes on Shimura varieties with parahoric level structure
Abstract: We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. We also verify most of the He-Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are non-empty for these models. | math | 1 |
Title: Effective Ways to Build and Evaluate Individual Survival Distributions
Abstract: An accurate model of a patient's individual survival distribution can help determine the appropriate treatment for terminal patients. Unfortunately, risk scores (e.g., from Cox Proportional Hazard models) do not provide survival probabilities, single-time probability models (e.g., the Gail model, predicting 5 year probability) only provide for a single time point, and standard Kaplan-Meier survival curves provide only population averages for a large class of patients meaning they are not specific to individual patients. This motivates an alternative class of tools that can learn a model which provides an individual survival distribution which gives survival probabilities across all times - such as extensions to the Cox model, Accelerated Failure Time, an extension to Random Survival Forests, and Multi-Task Logistic Regression. This paper first motivates such "individual survival distribution" (ISD) models, and explains how they differ from standard models. It then discusses ways to evaluate such models - namely Concordance, 1-Calibration, Brier score, and various versions of L1-loss - and then motivates and defines a novel approach "D-Calibration", which determines whether a model's probability estimates are meaningful. We also discuss how these measures differ, and use them to evaluate several ISD prediction tools, over a range of survival datasets. | cs | 1 |
Title: Fourier quasicrystals with unit masses
Abstract: Every set $\Lambda\subset R$ such that the sum of $\delta$-measures sitting at the points of $\Lambda$ is a Fourier quasicrystal, is the zero set of an exponential polynomial with imaginary frequencies. | math | 1 |
Title: Properly Outer and Strictly Outer Actions of Finite Groups on Prime C*-algebras
Abstract: An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper I define the notion of strictly outer action (similar to the definition for von Neumann factors in [11]) and prove that for finite groups it is equivalent with proper outerness of the action. For finite abelian groups this is equivalent with other relevant properities of the action. | math | 0 |
Title: Using Locality-sensitive Hashing for Rendezvous Search
Abstract: The multichannel rendezvous problem is a fundamental problem for neighbor discovery in many IoT applications. The existing works in the literature focus mostly on improving the worst-case performance, and the average-case performance is often not as good as that of the random algorithm. As IoT devices (users) are close to each other, their available channel sets, though they might be different, are similar. Using the locality-sensitive hashing (LSH) technique in data mining, we propose channel hopping algorithms that exploit the similarity between the two available channel sets to increase the rendezvous probability. For the synchronous setting, our algorithms have the expected time-to-rendezvous (ETTR) inversely proportional to a well-known similarity measure called the Jaccard index. For the asynchronous setting, we use dimensionality reduction to speed up the rendezvous process. Our numerical results show that our algorithms can outperform the random algorithm in terms of ETTR. | cs | 1 |
Title: Politics and Propaganda on Social Media: How Twitter and Meta Moderate State-Linked Information Operations
Abstract: Why do Social Media Corporations (SMCs) engage in state-linked information operations? Social media can significantly influence the global political landscape, allowing governments and other political entities to engage in concerted information operations, shaping or manipulating domestic and foreign political agendas. In response to state-linked political manipulation tactics on social media, Twitter and Meta carried out take-down operations against propaganda networks, accusing them of interfering foreign elections, organizing disinformation campaigns, manipulating political debates and many other issues. This research investigates the two SMCs' policy orientation to explain which factors can affect these two companies' reaction against state-linked information operations. We find that good governance indicators such as democracy are significant elements of SMCs' country-focus. This article also examines whether Meta and Twitter's attention to political regime characteristics is influenced by international political alignments. This research illuminates recent trends in SMCs' take-down operations and illuminating interplay between geopolitics and domestic regime characteristics. | cs | 0 |
Title: On Language Varieties Without Boolean Operations
Abstract: Eilenberg's variety theorem marked a milestone in the algebraic theory of regular languages by establishing a formal correspondence between properties of regular languages and properties of finite monoids recognizing them. Motivated by classes of languages accepted by quantum finite automata, we introduce basic varieties of regular languages, a weakening of Eilenberg's original concept that does not require closure under any boolean operations, and prove a variety theorem for them. To do so, we investigate the algebraic recognition of languages by lattice bimodules, generalizing Klima and Polak's lattice algebras, and we utilize the duality between algebraic completely distributive lattices and posets. | cs | 1 |
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. | cs | 0 |
Title: Learning Compact Reward for Image Captioning
Abstract: Adversarial learning has shown its advances in generating natural and diverse descriptions in image captioning. However, the learned reward of existing adversarial methods is vague and ill-defined due to the reward ambiguity problem. In this paper, we propose a refined Adversarial Inverse Reinforcement Learning (rAIRL) method to handle the reward ambiguity problem by disentangling reward for each word in a sentence, as well as achieve stable adversarial training by refining the loss function to shift the generator towards Nash equilibrium. In addition, we introduce a conditional term in the loss function to mitigate mode collapse and to increase the diversity of the generated descriptions. Our experiments on MS COCO and Flickr30K show that our method can learn compact reward for image captioning. | cs | 1 |
Title: Topological generation and matrix models for quantum reflection groups
Abstract: We establish several new topological generation results for the quantum permutation groups $S^+_N$ and the quantum reflection groups $H^{s+}_N$. We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for $S_N^+$ are inner faithful. | math | 1 |
Title: Structure and Design of HoloGen
Abstract: Increasing popularity of augmented and mixed reality systems has seen a similar increase of interest in 2D and 3D computer generated holography (CGH). Unlike stereoscopic approaches, CGH can fully represent a light field including depth of focus, accommodation and vergence. Along with existing telecommunications, imaging, projection, lithography, beam shaping and optical tweezing applications, CGH is an exciting technique applicable to a wide array of photonic problems including full 3D representation. Traditionally, the primary roadblock to acceptance has been the significant numerical processing required to generate holograms requiring both significant expertise and significant computational power. This article discusses the structure and design of HoloGen. HoloGen is an MIT licensed application that may be used to generate holograms using a wide array of algorithms without expert guidance. HoloGen uses a Cuda C and C++ backend with a C# and Windows Presentation Framework graphical user interface. The article begins by introducing HoloGen before providing an in-depth discussion of its design and structure. Particular focus is given to the communication, data transfer and algorithmic aspects. | cs | 1 |
Title: Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows
Abstract: We provide an exposition of a `horizontal' generalization of Goodman's surgery operation on (pseudo-)Anosov flows. This operation is performed by cutting along a specific kind of annulus that is transverse to the flow and regluing with a Dehn twist of the appropriate sign. We then show that performing horizontal Goodman surgery on a transitive pseudo-Anosov flow yields an almost equivalent flow, i.e. the original flow and the surgered flow are orbit equivalent after drilling out a finite collection of closed orbits. We obtain some almost equivalence results by applying this theorem on examples of the surgery operation. Along the way, we also show a structural stability result for pseudo-Anosov flows. | math | 0 |
Title: KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems
Abstract: We prove that the Cuntz-Pimsner algebra of every Temperley-Lieb subproduct system is KK-self-dual. We show also that every such Cuntz-Pimsner algebra has a canonical KMS-state, which we use to construct a Fredholm module representative for the fundamental class of the duality. This allows us to describe the K-homology of the Cuntz-Pimsner algebras by explicit Fredholm modules. Both the construction of the dual class and the proof of duality rely in a crucial way on quantum symmetries of Temperley-Lieb subproduct systems. In the simplest case of Arveson's $2$-shift our work establishes $U(2)$-equivariant KK-self-duality of $S^3$. | math | 0 |
Title: An experimental sorting method for improving metagenomic data encoding
Abstract: Minimizing data storage poses a significant challenge in large-scale metagenomic projects. In this paper, we present a new method for improving the encoding of FASTQ files generated by metagenomic sequencing. This method incorporates metagenomic classification followed by a recursive filter for clustering reads by DNA sequence similarity to improve the overall reference-free compression. In the results, we show an overall improvement in the compression of several datasets. As hypothesized, we show a progressive compression gain for higher coverage depth and number of identified species. Additionally, we provide an implementation that is freely available at https://github.com/cobilab/mizar and can be customized to work with other FASTQ compression tools. | cs | 0 |
Title: The quasi-static plasmonic problem for polyhedra
Abstract: We characterize the essential spectrum of the plasmonic problem for polyhedra in $\mathbb{R}^3$. The description is particularly simple for convex polyhedra and permittivities $\epsilon < - 1$. The plasmonic problem is interpreted as a spectral problem through a boundary integral operator, the direct value of the double layer potential, also known as the Neumann--Poincar\'e operator. We therefore study the spectral structure of the the double layer potential for polyhedral cones and polyhedra. | math | 1 |
Title: TR-DETR: Task-Reciprocal Transformer for Joint Moment Retrieval and Highlight Detection
Abstract: Video moment retrieval (MR) and highlight detection (HD) based on natural language queries are two highly related tasks, which aim to obtain relevant moments within videos and highlight scores of each video clip. Recently, several methods have been devoted to building DETR-based networks to solve both MR and HD jointly. These methods simply add two separate task heads after multi-modal feature extraction and feature interaction, achieving good performance. Nevertheless, these approaches underutilize the reciprocal relationship between two tasks. In this paper, we propose a task-reciprocal transformer based on DETR (TR-DETR) that focuses on exploring the inherent reciprocity between MR and HD. Specifically, a local-global multi-modal alignment module is first built to align features from diverse modalities into a shared latent space. Subsequently, a visual feature refinement is designed to eliminate query-irrelevant information from visual features for modal interaction. Finally, a task cooperation module is constructed to refine the retrieval pipeline and the highlight score prediction process by utilizing the reciprocity between MR and HD. Comprehensive experiments on QVHighlights, Charades-STA and TVSum datasets demonstrate that TR-DETR outperforms existing state-of-the-art methods. Codes are available at \url{https://github.com/mingyao1120/TR-DETR}. | cs | 0 |
Title: Bow varieties as symplectic reductions of $T^*(G/P)$
Abstract: Cherkis bow varieties were introduced as ADHM type description of moduli space of instantons on the Taub-NUT space equivariant under a cyclic group action. They are also models of Coulomb branches of quiver gauge theories of affine type A. In this paper, we realize each bow variety with torus fixed points as a symplectic reduction of a cotangent bundle of a partial flag variety by a unipotent group, and find a slice of this action. By this description, we calculate the equivariant cohomology (and ordinary cohomology) of some of them and answer some questions raisedbefore. This also uses a new result about circle-equivariant cohomology proven in an appendix. We also give an explicit generalized Mirkovic-Vybornov isomorphism for bow varieties in the appendix. | math | 0 |
Title: Gradient-Based Optimization of Lattice Quantizers
Abstract: Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension. | cs | 0 |
Title: Regularity for Maxwell eigenproblems in photonic crystal fibre modelling
Abstract: The convergence behaviour and the design of numerical methods for modelling the flow of light in photonic crystal fibres depend critically on an understanding of the regularity of solutions to time-harmonic Maxwell equations in a three-dimensional, periodic, translationally invariant, heterogeneous medium. In this paper we determine the strength of the dominant singularities that occur at the interface between materials. By modifying earlier regularity theory for polygonal interfaces we find that on each subdomain, where the material in the fibre is constant, the regularity of in-plane components of the magnetic field are $H^{2-\eta}$ for all $\eta > 0$. This estimate is sharp in the sense that these components do not belong to $H^2$, in general. However, global regularity is restricted by the presence of an interface between these subdomains and the interface conditions imply only $H^{3/2-\eta}$ regularity across the interface. The results are useful to anyone applying a numerical method such as a finite element method or a planewave expansion method to model photonic crystal fibres or similar materials. | math | 1 |
Title: Deplatforming Norm-Violating Influencers on Social Media Reduces Overall Online Attention Toward Them
Abstract: From politicians to podcast hosts, online platforms have systematically banned (``deplatformed'') influential users for breaking platform guidelines. Previous inquiries on the effectiveness of this intervention are inconclusive because 1) they consider only few deplatforming events; 2) they consider only overt engagement traces (e.g., likes and posts) but not passive engagement (e.g., views); 3) they do not consider all the potential places users impacted by the deplatforming event might migrate to. We address these limitations in a longitudinal, quasi-experimental study of 165 deplatforming events targeted at 101 influencers. We collect deplatforming events from Reddit posts and then manually curate the data, ensuring the correctness of a large dataset of deplatforming events. Then, we link these events to Google Trends and Wikipedia page views, platform-agnostic measures of online attention that capture the general public's interest in specific influencers. Through a difference-in-differences approach, we find that deplatforming reduces online attention toward influencers. After 12 months, we estimate that online attention toward deplatformed influencers is reduced by -63% (95% CI [-75%,-46%]) on Google and by -43% (95% CI [-57%,-24%]) on Wikipedia. Further, as we study over a hundred deplatforming events, we can analyze in which cases deplatforming is more or less impactful, revealing nuances about the intervention. Notably, we find that both permanent and temporary deplatforming reduce online attention toward influencers; Overall, this work contributes to the ongoing effort to map the effectiveness of content moderation interventions, driving platform governance away from speculation. | cs | 0 |
Title: Robust Regret Optimal Control
Abstract: This paper presents a synthesis method for robust, regret optimal control. The plant is modeled in discrete-time by an uncertain linear time-invariant (LTI) system. An optimal non-causal controller is constructed using the nominal plant model and given full knowledge of the disturbance. Robust regret is defined relative to the performance of this optimal non-causal control. It is shown that a controller achieves robust regret if and only if it satisfies a robust $H_\infty$ performance condition. DK-iteration can be used to synthesize a controller that satisfies this condition and hence achieve a given level of robust regret. The approach is demonstrated three examples: (i) a simple single-input, single-output classical design, (ii) a longitudinal control for a simplified model for a Boeing 747 model, and (iii) an active suspension for a quarter car model. All examples compare the robust regret optimal against regret optimal controllers designed without uncertainty. | math | 0 |
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. | cs | 0 |
Title: Robust and Adaptive Planning under Model Uncertainty
Abstract: Planning under model uncertainty is a fundamental problem across many applications of decision making and learning. In this paper, we propose the Robust Adaptive Monte Carlo Planning (RAMCP) algorithm, which allows computation of risk-sensitive Bayes-adaptive policies that optimally trade off exploration, exploitation, and robustness. RAMCP formulates the risk-sensitive planning problem as a two-player zero-sum game, in which an adversary perturbs the agent's belief over the models. We introduce two versions of the RAMCP algorithm. The first, RAMCP-F, converges to an optimal risk-sensitive policy without having to rebuild the search tree as the underlying belief over models is perturbed. The second version, RAMCP-I, improves computational efficiency at the cost of losing theoretical guarantees, but is shown to yield empirical results comparable to RAMCP-F. RAMCP is demonstrated on an n-pull multi-armed bandit problem, as well as a patient treatment scenario. | cs | 1 |
Title: Characters of Representations of Quantum Groups of Type $A_n$
Abstract: We introduce the notion of characters of comodules over coribbon Hopf algebras. The case of quantum groups of type $A_n$ is studied. We establish a characteristic equation for the quantum matrix and a q-analogue of Harish-Chandra- Itzykson-Zuber integral | math | 1 |
Title: Connected sums and directed systems in knot Floer homologies
Abstract: We prove a number of fundamental properties about instanton knot Floer homology. Our arguments rely on general properties of sutured Floer theories and apply also in the Heegaard Floer and monopole Floer settings, where many of our results were already known. Our main result is the connected sum formula for instanton knot Floer homology. An extension of this result proves the oriented skein exact triangle for the minus version of instanton knot Floer homology. Finally, we derive a new model of the minus version of instanton knot Floer homology, which takes the form of a free, finitely generated chain complex over a polynomial ring, as opposed to a direct limit. This construction is new to all of the Floer theories. We explore these results also in the context of Heegaard Floer theory as well. | math | 0 |
Title: Solutions to complex $m$-Hessian type equation and its application
Abstract: In this paper, we introduce the class $\mathcal{E}_{m,F}(\Omega)$ and prove the existence of solutions of the complex $m-$Hessian type equation $-F(u(z),z)H_{m}(u)=\mu$ where $\mu$ vanishes on all of $m-$polar sets in the class $\mathcal{E}_{m,F}(\Omega).$ Next, we prove the existence of solutions of this equation in the class $\mathcal{E}_{m,F}(\Omega)$ if there exists subsolution in this class. Using the above results, we study subextension in the class $\mathcal{E}_{m,F}(\Omega).$ | math | 0 |
Title: An Example of Evolutionary Computation + Large Language Model Beating Human: Design of Efficient Guided Local Search
Abstract: It is often very tedious for human experts to design efficient algorithms. Recently, we have proposed a novel Algorithm Evolution using Large Language Model (AEL) framework for automatic algorithm design. AEL combines the power of a large language model and the paradigm of evolutionary computation to design, combine, and modify algorithms automatically. In this paper, we use AEL to design the guide algorithm for guided local search (GLS) to solve the well-known traveling salesman problem (TSP). AEL automatically evolves elite GLS algorithms in two days, with minimal human effort and no model training. Experimental results on 1,000 TSP20-TSP100 instances and TSPLib instances show that AEL-designed GLS outperforms state-of-the-art human-designed GLS with the same iteration budget. It achieves a 0% gap on TSP20 and TSP50 and a 0.032% gap on TSP100 in 1,000 iterations. Our findings mark the emergence of a new era in automatic algorithm design. | cs | 0 |
Title: Representation stability in the level 4 braid group
Abstract: We investigate the cohomology of the level 4 subgroup of the braid group, namely, the kernel of the mod 4 reduction of the Burau representation at $t=-1$. This group is also equal to the kernel of the mod 2 abelianization of the pure braid group. We give an exact formula for the first Betti number; it is a quartic polynomial in the number of strands. We also show that, like the pure braid group, the first homology satisfies uniform representation stability in the sense of Church and Farb. Unlike the pure braid group, the group of symmetries - the quotient of the braid group by the level 4 subgroup - is one for which the representation theory has not been well studied; we develop its representation theory. This group is a non-split extension of the symmetric group. As applications of our main results, we show that the rational cohomology ring of the level 4 braid group is not generated in degree 1 when the number of strands is at least 15, and we compute all Betti numbers of the level 4 braid group when the number of strands is at most 4. We also derive a new lower bound on the first rational Betti number of the hyperelliptic Torelli group and on the top rational Betti number of the level 4 mapping class group in genus 2. Finally, we apply our results to locate all of the 2-torsion points on the characteristic varieties of the pure braid group. | math | 1 |
Title: Randomly coupled differential equations with elliptic correlations
Abstract: We consider the long time asymptotic behavior of a large system of $N$ linear differential equations with random coefficients. We allow for general elliptic correlation structures among the coefficients, thus we substantially generalize our previous work [14] that was restricted to the independent case. In particular, we analyze a recent model in the theory of neural networks [27] that specifically focused on the effect of the distributional asymmetry in the random connectivity matrix $X$. We rigorously prove and slightly correct the explicit formula from [28] on the time decay as a function of the asymmetry parameter. Our main tool is an asymptotically precise formula for the normalized trace of $f(X) g(X^*)$, in the large $N$ limit, where $f$ and $g$ are analytic functions. | math | 1 |
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. | cs | 1 |
Title: Offset Hypersurfaces and Persistent Homology of Algebraic Varieties
Abstract: In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and algebraic optimization. Namely, we express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean Distance Degree of the starting variety, obtaining a new way to compute these degrees. Finally, we describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology. | math | 1 |
Title: Roots of crosscap slides and crosscap transpositions
Abstract: Let $N_{g}$ denote a closed nonorientable surface of genus $g$. For $g \geq 2$ the mapping class group $\mathcal{M}(N_{g})$ is generated by Dehn twists and one crosscap slide ($Y$-homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists have nontrivial roots. We give necessary and sufficient conditions for the existence of a root of a crosscap slide and a crosscap transposition. | math | 1 |
Title: Higher dimensional Calabi-Yau manifolds of Kummer type
Abstract: Based on Cynk-Hulek method we construct complex Calabi-Yau varieties of arbitrary dimensions using elliptic curves with automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize result of Katsura and Sch\"utt to obtain arbitrarily dimensional Calabi-Yau manifolds which are Zariski in any characteristic $p\not\equiv 1\pmod{12}.$ | math | 1 |
Title: Shifted Composition II: Shift Harnack Inequalities and Curvature Upper Bounds
Abstract: We apply the shifted composition rule -- an information-theoretic principle introduced in our earlier work [AC23] -- to establish shift Harnack inequalities for the Langevin diffusion. We obtain sharp constants for these inequalities for the first time, allowing us to investigate their relationship with other properties of the diffusion. Namely, we show that they are equivalent to a sharp "local gradient-entropy" bound, and that they imply curvature upper bounds in a compelling reflection of the Bakry-Emery theory of curvature lower bounds. Finally, we show that the local gradient-entropy inequality implies optimal concentration of the score, a.k.a. the logarithmic gradient of the density. | math | 0 |
Title: Convergence of boundary layers of chemotaxis models with physical boundary conditions~I: degenerate initial data
Abstract: The celebrated experiment of Tuval et al. \cite{tuval2005bacterial} showed that the bacteria living a water drop can form a thin layer near the air-water interface, where a so-called chemotaxis-fluid system with physical boundary conditions was proposed to interpret the mechanism underlying the pattern formation alongside numerical simulations. However, the rigorous proof for the existence and convergence of the boundary layer solutions to the proposed model still remains open. This paper shows that the model with physical boundary conditions proposed in \cite{tuval2005bacterial} in one dimension can generate boundary layer solution as the oxygen diffusion rate $\varepsilon>0$ is small. Specifically, we show that the solution of the model with $\varepsilon>0$ will converge to the solution with $\varepsilon=0$ (outer-layer solution) plus the boundary layer profiles (inner-layer solution) with a sharp transition near the boundary as $ \varepsilon \rightarrow 0$. There are two major difficulties in our analysis. First, the global well-posedness of the model is hard to prove since the Dirichlet boundary condition can not contribute to the gradient estimates needed for the cross-diffusion structure in the model. Resorting to the technique of taking anti-derivative, we remove the cross-diffusion structure such that the Dirichlet boundary condition can facilitate the needed estimates. Second, the outer-layer profile of bacterial density is required to be degenerate at the boundary as $ t \rightarrow 0 ^{+}$, which makes the traditional cancellation technique incapable. Here we employ the Hardy inequality and delicate weighted energy estimates to overcome this obstacle and derive the requisite uniform-in-$\varepsilon$ estimates allowing us to pass the limit $\varepsilon \to 0$ to achieve our results. | math | 0 |
Title: Gromov's Oka principle, fiber bundles and the conformal module
Abstract: The conformal module of conjugacy classes of braids is an invariant that appeared earlier than the entropy of conjugacy classes of braids, and is inverse proportional to the entropy. Using the relation between the two invariants we give a short conceptional proof of an earlier result on the conformal module. Mainly, we consider situations, when the conformal module of conjugacy classes of braids serves as obstruction for the existence of homotopies (or isotopies) of smooth objects involving braids to the respective holomorphic objects, and present theorems on the restricted validity of Gromov's Oka principle in these situations. | math | 1 |
Title: Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings
Abstract: We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods. | cs | 0 |
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. | cs | 1 |
Title: VSFormer: Visual-Spatial Fusion Transformer for Correspondence Pruning
Abstract: Correspondence pruning aims to find correct matches (inliers) from an initial set of putative correspondences, which is a fundamental task for many applications. The process of finding is challenging, given the varying inlier ratios between scenes/image pairs due to significant visual differences. However, the performance of the existing methods is usually limited by the problem of lacking visual cues (\eg texture, illumination, structure) of scenes. In this paper, we propose a Visual-Spatial Fusion Transformer (VSFormer) to identify inliers and recover camera poses accurately. Firstly, we obtain highly abstract visual cues of a scene with the cross attention between local features of two-view images. Then, we model these visual cues and correspondences by a joint visual-spatial fusion module, simultaneously embedding visual cues into correspondences for pruning. Additionally, to mine the consistency of correspondences, we also design a novel module that combines the KNN-based graph and the transformer, effectively capturing both local and global contexts. Extensive experiments have demonstrated that the proposed VSFormer outperforms state-of-the-art methods on outdoor and indoor benchmarks. Our code is provided at the following repository: https://github.com/sugar-fly/VSFormer. | cs | 0 |
Title: LaDe: The First Comprehensive Last-mile Delivery Dataset from Industry
Abstract: Real-world last-mile delivery datasets are crucial for research in logistics, supply chain management, and spatio-temporal data mining. Despite a plethora of algorithms developed to date, no widely accepted, publicly available last-mile delivery dataset exists to support research in this field. In this paper, we introduce \texttt{LaDe}, the first publicly available last-mile delivery dataset with millions of packages from the industry. LaDe has three unique characteristics: (1) Large-scale. It involves 10,677k packages of 21k couriers over 6 months of real-world operation. (2) Comprehensive information. It offers original package information, such as its location and time requirements, as well as task-event information, which records when and where the courier is while events such as task-accept and task-finish events happen. (3) Diversity. The dataset includes data from various scenarios, including package pick-up and delivery, and from multiple cities, each with its unique spatio-temporal patterns due to their distinct characteristics such as populations. We verify LaDe on three tasks by running several classical baseline models per task. We believe that the large-scale, comprehensive, diverse feature of LaDe can offer unparalleled opportunities to researchers in the supply chain community, data mining community, and beyond. The dataset homepage is publicly available at https://huggingface.co/datasets/Cainiao-AI/LaDe. | cs | 0 |
Title: 3-anti-power uniform morphisms
Abstract: Words whose three successive factors of the same length are all different i.e. 3-anti-power words are a natural extension of square-free words (two successive factors of the same length are different). We give a way to verify whether a uniform morphism preserves 3-anti-power words (the image of a 3-anti-power word is a 3-anti-power word). A consequence of the existence of such morphisms is the possibility of generating an infinite 3-anti-power word. | cs | 0 |
Title: Ramified covering maps of singular curves and stability of pulled back bundles
Abstract: Let $f : X \rightarrow Y$ be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is \'etale on an open subset of $Y$ that contains both the singular locus of $Y$ and the image, in $Y$, of the singular locus of $X$. We prove that the following statements are equivalent: \begin{enumerate} \item The homomorphism of \'etale fundamental groups $$f_* : \pi_1^{\rm et}(X) \rightarrow\pi_1^{\rm et}(Y)$$ induced by $f$ is surjective. \item There is no nontrivial \'etale covering $\phi : Y' \rightarrow Y$ admitting a morphism $q: X \rightarrow Y'$ such that $\phi\circ q = f$. \item The fiber product $X\times_Y X$ is connected. \item $\dim H^0(X, f^*f_* {\mathcal O}_X)= 1$. \item ${\mathcal O}_Y \subset f_*{\mathcal O}_X$ is the maximal semistable subsheaf. \item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate} | math | 0 |
Title: The Adjoint Representation of a Higher Lie Groupoid
Abstract: We extend the standard construction of the adjoint representation of a Lie groupoid to the case of an arbitrary higher Lie groupoid. As for a Lie groupoid, the adjoint representation of a higher Lie groupoid turns out to be a representation up to homotopy which is well defined up to isomorphism. Its existence and uniqueness are immediate consequences of a more general result in the theory of simplicial vector bundles: the representation up to homotopy obtained by splitting a higher vector bundle by means of a cleavage is, to within isomorphism, independent of the choice of the cleavage. | math | 0 |
Title: On Nontrivial Winning and Losing Parameters of Schmidt Games
Abstract: In this paper we completely describe the winning and losing conditions different from the only ``trivial'' conditions known before. In other words, we solve the open question of finding a complete nontrivial Schmidt diagram. In addition, we give the new bounds for two family of sets: one related to frequencies of digits in base-$2$ expansions, and one connected to the set of the badly approximable numbers. | math | 0 |
Title: SuperEdge: Towards a Generalization Model for Self-Supervised Edge Detection
Abstract: Edge detection is a fundamental technique in various computer vision tasks. Edges are indeed effectively delineated by pixel discontinuity and can offer reliable structural information even in textureless areas. State-of-the-art heavily relies on pixel-wise annotations, which are labor-intensive and subject to inconsistencies when acquired manually. In this work, we propose a novel self-supervised approach for edge detection that employs a multi-level, multi-homography technique to transfer annotations from synthetic to real-world datasets. To fully leverage the generated edge annotations, we developed SuperEdge, a streamlined yet efficient model capable of concurrently extracting edges at pixel-level and object-level granularity. Thanks to self-supervised training, our method eliminates the dependency on manual annotated edge labels, thereby enhancing its generalizability across diverse datasets. Comparative evaluations reveal that SuperEdge advances edge detection, demonstrating improvements of 4.9% in ODS and 3.3% in OIS over the existing STEdge method on BIPEDv2. | cs | 0 |
Title: Investigating the Suitability of Concept Drift Detection for Detecting Leakages in Water Distribution Networks
Abstract: Leakages are a major risk in water distribution networks as they cause water loss and increase contamination risks. Leakage detection is a difficult task due to the complex dynamics of water distribution networks. In particular, small leakages are hard to detect. From a machine-learning perspective, leakages can be modeled as concept drift. Thus, a wide variety of drift detection schemes seems to be a suitable choice for detecting leakages. In this work, we explore the potential of model-loss-based and distribution-based drift detection methods to tackle leakage detection. We additionally discuss the issue of temporal dependencies in the data and propose a way to cope with it when applying distribution-based detection. We evaluate different methods systematically for leakages of different sizes and detection times. Additionally, we propose a first drift-detection-based technique for localizing leakages. | cs | 0 |
Title: Design and Actuator Optimization of Lightweight and Compliant Knee Exoskeleton for Mobility Assistance of Children with Crouch Gait
Abstract: Pediatric exoskeletons offer great promise to increase mobility for children with crouch gait caused by cerebral palsy. A lightweight, compliant and user-specific actuator is critical for maximizing the benefits of an exoskeleton to users. To date, pediatric exoskeletons generally use the same actuators as adult exoskeletons, which are heavy and resistive to natural movement. There is yet no easy way for robotic exoskeletons to accommodate the changes in design requirements that occur as a child ages. We developed a lightweight (1.65 kg unilateral mass) and compliant pediatric knee exoskeleton with a bandwidth of 22.6 Hz that can provide torque assistance to children with crouch gait using high torque density motor. Experimental results demonstrated that the robot exhibited low mechanical impedance (1.79 Nm average backdrive torque) under the unpowered condition and 0.32 Nm with zero-torque tracking control. Root mean square (RMS) error of torque tracking result is less than 0.73 Nm (5.7% with respect to 12 Nm torque). To achieve optimal age-specific performance, we proposed the first optimization framework that considered both motor and transmission of the actuator system that can produce optimal settings for children between 3 and 18 years old. The optimization generated an optimal motor air gap radius that monotonically increases with age from 0.011 to 0.033 meters, and optimal gear ratio varies from 2.6 to 11.6 (3-13 years old) and 11.6 to 10.2 (13-18 years old), leading to actuators of minimal mass. | cs | 1 |
Title: Clairvoyant embedding in one dimension
Abstract: Let v, w be infinite 0-1 sequences, and m a positive integer. We say that w is m-embeddable in v, if there exists an increasing sequence n_{i} of integers with n_{0}=0, such that 0< n_{i} - n_{i-1} < m, w(i) = v(n_i) for all i > 0. Let X and Y be independent coin-tossing sequences. We will show that there is an m with the property that Y is m-embeddable into X with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation. | math | 1 |
Title: Learning circuits with few negations
Abstract: Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A. A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions). | cs | 1 |
Title: Affine homogeneous varieties and suspensions
Abstract: An algebraic variety $X$ is called a homogeneous variety if the automorphism group $\mathrm{Aut}(X)$ acts on $X$ transitively, and a homogeneous space if there exists a transitive action of an algebraic group on $X$. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces. | math | 0 |
Title: A comparison of the Spectral Ewald and Smooth Particle Mesh Ewald methods in GROMACS
Abstract: The smooth particle mesh Ewald (SPME) method is an FFT based method for the fast evaluation of electrostatic interactions under periodic boundary conditions. A highly optimized implementation of this method is available in GROMACS, a widely used software for molecular dynamics simulations. In this article, we compare a more recent method from the same family of methods, the spectral Ewald (SE) method, to the SPME method in terms of performance and efficiency. We consider serial and parallel implementations of both methods for single and multiple core computations on a desktop machine as well as the Beskow supercomputer at KTH Royal Institute of Technology. The implementation of the SE method has been well optimized, however not yet comparable to the level of the SPME implementation that has been improved upon for many years. We show that the SE method is very efficient whenever used to achieve high accuracy and that it already at this level of optimization can be competitive for low accuracy demands. | math | 1 |
Title: tmf Is Not a Ring Spectrum Quotient of String Bordism
Abstract: This paper shows that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}\langle8\rangle[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of $\pi_*\mathrm{MO}\langle8\rangle$, the $\pi_*\mathrm{MO}\langle8\rangle$-module $$\pi_*\big(\mathrm{MO}\langle8\rangle_{(p)}/X\big)$$ is not (even abstractly) isomorphic to $\pi_*\mathrm{tmf}_{(p)}$. It does so by showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of $\pi_*(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces $$\pi_*(R/X)\otimes\mathbf{Q} \cong \mathrm{H}_*(\mathrm{Tot}(\mathrm{K}(X)))\otimes\mathbf{Q},$$ where the right-hand side is the rational homology of the (total) Koszul complex of $X$, which is strictly bigger than $\pi_*(R)/(X)\otimes\mathbf{Q}$ unless $X$ is a $\pi_*(R)\otimes\mathbf{Q}$-quasi-regular sequence. The result then follows from the fact that the kernel of the $p$-local Witten genus cannot be generated by a $\pi_*\mathrm{MO}\langle8\rangle\otimes\mathbf{Q}$-quasi-regular sequence. | math | 1 |
Title: A note on minor antichains of uncountable graphs
Abstract: A simplified construction is presented for Komj\'ath's result that for every uncountable cardinal $\kappa$, there are $2^\kappa$ graphs of size $\kappa$ none of them being a minor of another. | math | 1 |
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. | math | 0 |
Title: The Constrained Round Robin Algorithm for Fair and Efficient Allocation
Abstract: We consider a multi-agent resource allocation setting that models the assignment of papers to reviewers. A recurring issue in allocation problems is the compatibility of welfare/efficiency and fairness. Given an oracle to find a welfare-achieving allocation, we embed such an oracle into a flexible algorithm called the Constrained Round Robin (CRR) algorithm, that achieves the required welfare level. Our algorithm also allows the system designer to lower the welfare requirements in order to achieve a higher degree of fairness. If the welfare requirement is lowered enough, a strengthening of envy-freeness up to one item is guaranteed. Hence, our algorithm can be viewed as a computationally efficient way to interpolate between welfare and approximate envy-freeness in allocation problems. | cs | 1 |
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. | math | 0 |
Title: Phenotype switching in chemotaxis aggregation models controls the spontaneous emergence of large densities
Abstract: We consider a phenotype-switching chemotaxis model for aggregation, in which a chemotactic population is capable of switching back and forth between a chemotaxing state (performing chemotactic movement) and a secreting state (producing the attractant). We show that the switching rate provides a powerful mechanism for controlling the densities of spontaneously emerging aggregates. Specifically, in two- and three-dimensional settings it is shown that when both switching rates coincide and are suitably large, then the densities of both the chemotaxing and the secreting population will exceed any prescribed level at some points in the considered domain. This is complemented by two results asserting the absence of such aggregation phenomena in corresponding scenarios in which one of the switching rates remains within some bounded interval. | math | 1 |
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 $q$. | cs | 0 |
Title: CAD-compatible structural shape optimization with a movable Bézier tetrahedral mesh
Abstract: This paper presents the development of a complete CAD-compatible framework for structural shape optimization in 3D. The boundaries of the domain are described using NURBS while the interior is discretized with B\'ezier tetrahedra. The tetrahedral mesh is obtained from the mesh generator software Gmsh. A methodology to reconstruct the NURBS surfaces from the triangular faces of the boundary mesh is presented. The description of the boundary is used for the computation of the analytical sensitivities with respect to the control points employed in surface design. Further, the mesh is updated at each iteration of the structural optimization process by a pseudo-elastic moving mesh method. In this procedure, the existing mesh is deformed to match the updated surface and therefore reduces the need for remeshing. Numerical examples are presented to test the performance of the proposed method. The use of the movable mesh technique results in a considerable decrease in the computational effort for the numerical examples. | cs | 0 |
Title: Descent distribution on Catalan words avoiding a pattern of length at most three
Abstract: Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of descents on this set. Some of the obtained enumerating sequences are not yet recorded in the On-line Encyclopedia of Integer Sequences. | math | 1 |
Title: DEWP: Deep Expansion Learning for Wind Power Forecasting
Abstract: Wind is one kind of high-efficient, environmentally-friendly and cost-effective energy source. Wind power, as one of the largest renewable energy in the world, has been playing a more and more important role in supplying electricity. Though growing dramatically in recent years, the amount of generated wind power can be directly or latently affected by multiple uncertain factors, such as wind speed, wind direction, temperatures, etc. More importantly, there exist very complicated dependencies of the generated power on the latent composition of these multiple time-evolving variables, which are always ignored by existing works and thus largely hinder the prediction performances. To this end, we propose DEWP, a novel Deep Expansion learning for Wind Power forecasting framework to carefully model the complicated dependencies with adequate expressiveness. DEWP starts with a stack-by-stack architecture, where each stack is composed of (i) a variable expansion block that makes use of convolutional layers to capture dependencies among multiple variables; (ii) a time expansion block that applies Fourier series and backcast/forecast mechanism to learn temporal dependencies in sequential patterns. These two tailored blocks expand raw inputs into different latent feature spaces which can model different levels of dependencies of time-evolving sequential data. Moreover, we propose an inference block corresponding for each stack, which applies multi-head self-attentions to acquire attentive features and maps expanded latent representations into generated wind power. In addition, to make DEWP more expressive in handling deep neural architectures, we adapt doubly residue learning to process stack-by-stack outputs. Finally, we present extensive experiments in the real-world wind power forecasting application on two datasets from two different turbines to demonstrate the effectiveness of our approach. | cs | 0 |
Title: A quatum inspired neural network for geometric modeling
Abstract: By conceiving physical systems as 3D many-body point clouds, geometric graph neural networks (GNNs), such as SE(3)/E(3) equivalent GNNs, have showcased promising performance. In particular, their effective message-passing mechanics make them adept at modeling molecules and crystalline materials. However, current geometric GNNs only offer a mean-field approximation of the many-body system, encapsulated within two-body message passing, thus falling short in capturing intricate relationships within these geometric graphs. To address this limitation, tensor networks, widely employed by computational physics to handle manybody systems using high-order tensors, have been introduced. Nevertheless, integrating these tensorized networks into the message-passing framework of GNNs faces scalability and symmetry conservation (e.g., permutation and rotation) challenges. In response, we introduce an innovative equivariant Matrix Product State (MPS)-based message-passing strategy, through achieving an efficient implementation of the tensor contraction operation. Our method effectively models complex many-body relationships, suppressing mean-field approximations, and captures symmetries within geometric graphs. Importantly, it seamlessly replaces the standard message-passing and layer-aggregation modules intrinsic to geometric GNNs. We empirically validate the superior accuracy of our approach on benchmark tasks, including predicting classical Newton systems and quantum tensor Hamiltonian matrices. To our knowledge, our approach represents the inaugural utilization of parameterized geometric tensor networks. | cs | 0 |
Title: The colouring number of infinite graphs
Abstract: We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its vertices that simultaneously witnesses its colouring number and its cardinality. | math | 1 |
Title: Constant Step Size Least-Mean-Square: Bias-Variance Trade-offs and Optimal Sampling Distributions
Abstract: We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis leads to new insights into stochastic approximation algorithms: (a) it gives a tighter bound on the allowed step-size; (b) the generalization error may be divided into a variance term which is decaying as O(1/n), independently of the step-size $\gamma$, and a bias term that decays as O(1/$\gamma$ 2 n 2); (c) when allowing non-uniform sampling, the choice of a good sampling density depends on whether the variance or bias terms dominate. In particular, when the variance term dominates, optimal sampling densities do not lead to much gain, while when the bias term dominates, we can choose larger step-sizes that leads to significant improvements. | cs | 1 |
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. | math | 0 |
Title: An obstruction relating locally finite polygons to translation quadrangles
Abstract: One of the most fundamental open problems in Incidence Geometry, posed by Tits in the 1960s, asks for the existence of so-called "locally finite generalized polygons" | that is, generalized polygons with "mixed parameters" (one being finite and the other not). In a more specialized context, another long-standing problem (from the 1990s) is as to whether the endomorphism ring of any translation generalized quadrangle is a skew field (the answer of which is known in the finite case). (The analogous problem for projective planes, and its positive solution, the "Bruck-Bose construction," lies at the very base of the whole theory of translation planes.) In this short note, we introduce a category, representing certain very specific embeddings of generalized polygons, which surprisingly controls the solution of both (apparently entirely unrelated) problems. | math | 1 |
Title: Universal height and width bounds for random trees
Abstract: We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson (https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degree statistics, and a random truncation of a size-biased ordering of the degrees of such a tree. We also exploit a Poissonization trick introduced by Camarri and Pitman (https://doi.org/10.1214/EJP.v5-58) in the context of inhomogeneous continuum random trees, which we adapt to the setting of random trees with fixed degrees. Finally, we propose and justify a change to the conventions of branching process nomenclature: the name "Galton-Watson trees" should be permanently retired by the community, and replaced with the name "Bienaym\'e trees". | math | 1 |