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Title: Approximation of polynomials from Walsh tail spaces
Abstract: We derive various bounds for the $L_p$ distance of polynomials on the hypercube from Walsh tail spaces, extending some of Oleszkiewicz's results (2017) for Rademacher sums. | math | 0 |
Title: The (twisted/$L^2$)-Alexander polynomial of ideally triangulated 3-manifolds
Abstract: We establish a connection between the Alexander polynomial of a knot and its twisted and $L^2$-versions with the triangulations that appear in 3-dimensional hyperbolic geometry. Specifically, we introduce twisted Neumann--Zagier matrices of ordered ideal triangulations and use them to provide formulas for the Alexander polynomial and its variants, the twisted Alexander polynomial and the $L^2$-Alexander torsion. | math | 0 |
Title: Age-Aware Dynamic Frame Slotted ALOHA for Machine-Type Communications
Abstract: Information aging has gained prominence in characterizing communication protocols for timely remote estimation and control applications. This work proposes an Age of Information (AoI)-aware threshold-based dynamic frame slotted ALOHA (T-DFSA) for contention resolution in random access machine-type communication networks. Unlike conventional DFSA that maximizes the throughput in each frame, the frame length and age-gain threshold in T-DFSA are determined to minimize the normalized average AoI reduction of the network in each frame. At the start of each frame in the proposed protocol, the common Access Point (AP) stores an estimate of the age-gain distribution of a typical node. Depending on the observed status of the slots, age-gains of successful nodes, and maximum available AoI, the AP adjusts its estimation in each frame. The maximum available AoI is exploited to derive the maximum possible age-gain at each frame and thus, to avoid overestimating the age-gain threshold, which may render T-DFSA unstable. Numerical results validate our theoretical analysis and demonstrate the effectiveness of the proposed T-DFSA compared to the existing optimal frame slotted ALOHA, threshold-ALOHA, and age-based thinning protocols in a considerable range of update generation rates. | cs | 0 |
Title: A note on a Sung-Wang's paper
Abstract: The purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $\lambda_{1,p}$ for the $p$-Laplacian achievies its maximal value on a K\"{a}hler manifold or a quaternionic K\"{a}hler manifold then such a manifold must be connected at infinity unless it is a topological cylinder with an explicit warped product metric. | math | 1 |
Title: Derivative-Based Diagnosis of Regular Expression Ambiguity
Abstract: Regular expressions are often ambiguous. We present a novel method based on Brzozowski's derivatives to aid the user in diagnosing ambiguous regular expressions. We introduce a derivative-based finite state transducer to generate parse trees and minimal counter-examples. The transducer can be easily customized to either follow the POSIX or Greedy disambiguation policy and based on a finite set of examples it is possible to examine if there are any differences among POSIX and Greedy. | cs | 1 |
Title: Cartesian closed and stable subconstructs of Q-Ord
Abstract: It is shown that the category of fuzzy ordered sets and order-preserving maps valued in the quantale based on a continuous triangular norm on the unit interval contains a largest Cartesian closed and stable subconstruct which contains all crisp ordered sets. | math | 0 |
Title: Quantum rigidity of negatively curved manifolds
Abstract: We show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected Riemannian manifold $(M,g)$ with strictly negative curvature is automatically classical, in the sense that it factors through the action of the isometry group of $(M,g)$. This partially answers a question by D. Goswami. | math | 1 |
Title: Using Schur Rings to Produce GRRs for Dihedral Groups
Abstract: In this paper we shall be looking at several results relating Schur rings to sufficient conditions for a graph to be a graphical regular representation (GRR) of a finite group, and then applying these specifically in the case of certain subfamilies of dihedral groups. Numerical methods are given for constructing trivalent GRRs for these dihedral groups very quickly. | math | 0 |
Title: Investigating EEG-Based Functional Connectivity Patterns for Multimodal Emotion Recognition
Abstract: Compared with the rich studies on the motor brain-computer interface (BCI), the recently emerging affective BCI presents distinct challenges since the brain functional connectivity networks involving emotion are not well investigated. Previous studies on emotion recognition based on electroencephalography (EEG) signals mainly rely on single-channel-based feature extraction methods. In this paper, we propose a novel emotion-relevant critical subnetwork selection algorithm and investigate three EEG functional connectivity network features: strength, clustering coefficient, and eigenvector centrality. The discrimination ability of the EEG connectivity features in emotion recognition is evaluated on three public emotion EEG datasets: SEED, SEED-V, and DEAP. The strength feature achieves the best classification performance and outperforms the state-of-the-art differential entropy feature based on single-channel analysis. The experimental results reveal that distinct functional connectivity patterns are exhibited for the five emotions of disgust, fear, sadness, happiness, and neutrality. Furthermore, we construct a multimodal emotion recognition model by combining the functional connectivity features from EEG and the features from eye movements or physiological signals using deep canonical correlation analysis. The classification accuracies of multimodal emotion recognition are 95.08/6.42% on the SEED dataset, 84.51/5.11% on the SEED-V dataset, and 85.34/2.90% and 86.61/3.76% for arousal and valence on the DEAP dataset, respectively. The results demonstrate the complementary representation properties of the EEG connectivity features with eye movement data. In addition, we find that the brain networks constructed with 18 channels achieve comparable performance with that of the 62-channel network in multimodal emotion recognition and enable easier setups for BCI systems in real scenarios. | cs | 1 |
Title: Strongly Minimal Steiner Systems I: Existence
Abstract: A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner $k$-system (for $k \geq 2$) is a linear space such that each line has size exactly $k$. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary $\tau$ with a single ternary relation $R$. We prove that for every integer $k$ there exist $2^{\aleph_0}$-many integer valued functions $\mu$ such that each $\mu$ determines a distinct strongly minimal Steiner $k$-system $\mathcal{G}_\mu$, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture. | math | 1 |
Title: Learning to Generate Training Datasets for Robust Semantic Segmentation
Abstract: Semantic segmentation methods have advanced significantly. Still, their robustness to real-world perturbations and object types not seen during training remains a challenge, particularly in safety-critical applications. We propose a novel approach to improve the robustness of semantic segmentation techniques by leveraging the synergy between label-to-image generators and image-to-label segmentation models. Specifically, we design Robusta, a novel robust conditional generative adversarial network to generate realistic and plausible perturbed images that can be used to train reliable segmentation models. We conduct in-depth studies of the proposed generative model, assess the performance and robustness of the downstream segmentation network, and demonstrate that our approach can significantly enhance the robustness in the face of real-world perturbations, distribution shifts, and out-of-distribution samples. Our results suggest that this approach could be valuable in safety-critical applications, where the reliability of perception modules such as semantic segmentation is of utmost importance and comes with a limited computational budget in inference. We release our code at https://github.com/ENSTA-U2IS/robusta. | cs | 0 |
Title: Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems
Abstract: The homogenization of eigenvalues of non-Hermitian Maxwell operators is studied by the H-convergence method. It is assumed that the Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions of the form $n \times E = Z [(n \times H )\times n ] $. We show that, for a wide class of impedance operators $Z$, the nonzero spectrum of the corresponding Maxwell operator is discrete. To this end, a new continuous embedding theorem for domains of Maxwell operators is obtained. We prove the convergence of eigenvalues to an eigenvalue of a homogenized Maxwell operator under the assumption of the H-convergence of the material tensor-fields. This result is applied then to the existence of optimizers for eigenvalue optimization problems and to the existence of an eigenvalue-free region around zero. Connections with unique (and nonunique) continuation results are discussed. | math | 0 |
Title: Stochastic Analysis of an Adaptive Cubic Regularisation Method under Inexact Gradient Evaluations and Dynamic Hessian Accuracy
Abstract: We here adapt an extended version of the adaptive cubic regularisation method with dynamic inexact Hessian information for nonconvex optimisation in [3] to the stochastic optimisation setting. While exact function evaluations are still considered, this novel variant inherits the innovative use of adaptive accuracy requirements for Hessian approximations introduced in [3] and additionally employs inexact computations of the gradient. Without restrictions on the variance of the errors, we assume that these approximations are available within a sufficiently large, but fixed, probability and we extend, in the spirit of [18], the deterministic analysis of the framework to its stochastic counterpart, showing that the expected number of iterations to reach a first-order stationary point matches the well known worst-case optimal complexity. This is, in fact, still given by O(epsilon^(-3/2)), with respect to the first-order epsilon tolerance. Finally, numerical tests on nonconvex finite-sum minimisation confirm that using inexact first and second-order derivatives can be beneficial in terms of the computational savings. | math | 1 |
Title: SwitchTab: Switched Autoencoders Are Effective Tabular Learners
Abstract: Self-supervised representation learning methods have achieved significant success in computer vision and natural language processing, where data samples exhibit explicit spatial or semantic dependencies. However, applying these methods to tabular data is challenging due to the less pronounced dependencies among data samples. In this paper, we address this limitation by introducing SwitchTab, a novel self-supervised method specifically designed to capture latent dependencies in tabular data. SwitchTab leverages an asymmetric encoder-decoder framework to decouple mutual and salient features among data pairs, resulting in more representative embeddings. These embeddings, in turn, contribute to better decision boundaries and lead to improved results in downstream tasks. To validate the effectiveness of SwitchTab, we conduct extensive experiments across various domains involving tabular data. The results showcase superior performance in end-to-end prediction tasks with fine-tuning. Moreover, we demonstrate that pre-trained salient embeddings can be utilized as plug-and-play features to enhance the performance of various traditional classification methods (e.g., Logistic Regression, XGBoost, etc.). Lastly, we highlight the capability of SwitchTab to create explainable representations through visualization of decoupled mutual and salient features in the latent space. | cs | 0 |
Title: An Inversion Formula for the Gaussian Radon Transform for Banach Spaces
Abstract: We provide a disintegration theorem for the Gaussian Radon transform Gf on Banach spaces and use the Segal-Bargmann transform on abstract Wiener spaces to find a procedure to obtain f from its Gaussian Radon transform Gf. | math | 1 |
Title: Further Explanations on "SAT Requires Exhaustive Search"
Abstract: Recently, Xu and Zhou [2023] introduced a constructive approach for exploring computational hardness, proving that SAT requires exhaustive search. In light of certain misinterpretations concerning the contributions and proofs in that paper, we focus on providing detailed explanations in this work. We begin by delineating the core innovation of the constructive approach, shedding light on the pivotal concept of algorithm designability. We address the overlooked white-box diagonalization method and highlight the concept of an almost independent solution space. In response to specific misunderstandings, such as the concerns surrounding the assumptions of Lemma 3.1, we offer comprehensive clarifications aimed at improving the comprehension of the proof. We are grateful for the feedback received on our prior paper and hope this work can foster a more well-informed discussion. | cs | 0 |
Title: Complexity Classes and Completeness in Algebraic Geometry
Abstract: We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting. | math | 1 |
Title: Lower bounds for the eigenvalue estimates of the submanifold Dirac operator
Abstract: We get optimal lower bounds for the eigenvalues of the submanifold Dirac operator on locally reducible Riemannian manifolds in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied. As a corollary, one gets several known results in this direction. | math | 1 |
Title: Boltzmann equation with mixed boundary condition
Abstract: We study the Boltzmann equation in a smooth bounded domain featuring a mixed boundary condition. Specifically, gas particles experience specular reflection in two parallel plates, while diffusive reflection occurs in the remaining portion between these two specular regions. The boundary is assumed to be motionless and isothermal. Our main focus is on constructing global-in-time small-amplitude solutions around global Maxwellians for the corresponding initial-boundary value problem. The proof relies on the $L^2$ hypocoercivity at the linear level, utilizing the weak formulation and various functional inequalities on the test functions, such as Poincar\'e and Korn inequalities. It also extends to the linear problem involving Maxwell boundary conditions, where the accommodation coefficient can be a piecewise constant function on the boundary, allowing for more general bounded domains. Moreover, we develop a delicate application of the $L^2-L^\infty$ bootstrap argument, which relies on the specific geometry of our domains, to effectively handle this mixed-type boundary condition. | math | 0 |
Title: The triviality of a certain invariant of link maps in the four-sphere
Abstract: It is an open problem whether Kirk's $\sigma$ invariant is the complete obstruction to a link map $S^2\cup S^2\to S^4$ being link homotopically trivial. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $\omega$ that is defined on the kernel of $\sigma$ and also obstructs link nullhomotopy. We show that $\omega$ is determined by $\sigma$, and is a strictly weaker invariant. | math | 1 |
Title: Precondition and Effect Reasoning for Action Recognition
Abstract: Human action recognition has drawn a lot of attention in the recent years due to the research and application significance. Most existing works on action recognition focus on learning effective spatial-temporal features from videos, but neglect the strong causal relationship among the precondition, action and effect. Such relationships are also crucial to the accuracy of action recognition. In this paper, we propose to model the causal relationships based on the precondition and effect to improve the performance of action recognition. Specifically, a Cycle-Reasoning model is proposed to capture the causal relationships for action recognition. To this end, we annotate precondition and effect for a large-scale action dataset. Experimental results show that the proposed Cycle-Reasoning model can effectively reason about the precondition and effect and can enhance action recognition performance. | cs | 1 |
Title: A compactness result for the CR Yamabe problem in three dimensions
Abstract: We prove the compactness of the set of solutions to the CR Yamabe problem on a compact strictly pseudoconvex CR manifold of dimension three whose blow-up manifolds at every point have positive p-mass. As a corollary we deduce that compactness holds for CR-embeddable manifolds which are not CR-equivalent to $S^3$. The theorem is proved by blow-up analysis. | math | 0 |
Title: Software Testing, AI and Robotics (STAIR) Learning Lab
Abstract: In this paper we presented the Software Testing, AI and Robotics (STAIR) Learning Lab. STAIR is an initiative started at the University of Innsbruck to bring robotics, Artificial Intelligence (AI) and software testing into schools. In the lab physical and virtual learning units are developed in parallel and in sync with each other. Its core learning approach is based the develop of both a physical and simulated robotics environment. In both environments AI scenarios (like traffic sign recognition) are deployed and tested. We present and focus on our newly designed MiniBot that are both built on hardware which was designed for educational and research purposes as well as the simulation environment. Additionally, we describe first learning design concepts and a showcase scenario (i.e., AI-based traffic sign recognition) with different exercises which can easily be extended. | cs | 1 |
Title: From Merging Frameworks to Merging Stars: Experiences using HPX, Kokkos and SIMD Types
Abstract: Octo-Tiger, a large-scale 3D AMR code for the merger of stars, uses a combination of HPX, Kokkos and explicit SIMD types, aiming to achieve performance-portability for a broad range of heterogeneous hardware. However, on A64FX CPUs, we encountered several missing pieces, hindering performance by causing problems with the SIMD vectorization. Therefore, we add std::experimental::simd as an option to use in Octo-Tiger's Kokkos kernels alongside Kokkos SIMD, and further add a new SVE (Scalable Vector Extensions) SIMD backend. Additionally, we amend missing SIMD implementations in the Kokkos kernels within Octo-Tiger's hydro solver. We test our changes by running Octo-Tiger on three different CPUs: An A64FX, an Intel Icelake and an AMD EPYC CPU, evaluating SIMD speedup and node-level performance. We get a good SIMD speedup on the A64FX CPU, as well as noticeable speedups on the other two CPU platforms. However, we also experience a scaling issue on the EPYC CPU. | cs | 1 |
Title: DB-GPT: Empowering Database Interactions with Private Large Language Models
Abstract: The recent breakthroughs in large language models (LLMs) are positioned to transition many areas of software. Database technologies particularly have an important entanglement with LLMs as efficient and intuitive database interactions are paramount. In this paper, we present DB-GPT, a revolutionary and production-ready project that integrates LLMs with traditional database systems to enhance user experience and accessibility. DB-GPT is designed to understand natural language queries, provide context-aware responses, and generate complex SQL queries with high accuracy, making it an indispensable tool for users ranging from novice to expert. The core innovation in DB-GPT lies in its private LLM technology, which is fine-tuned on domain-specific corpora to maintain user privacy and ensure data security while offering the benefits of state-of-the-art LLMs. We detail the architecture of DB-GPT, which includes a novel retrieval augmented generation (RAG) knowledge system, an adaptive learning mechanism to continuously improve performance based on user feedback and a service-oriented multi-model framework (SMMF) with powerful data-driven agents. Our extensive experiments and user studies confirm that DB-GPT represents a paradigm shift in database interactions, offering a more natural, efficient, and secure way to engage with data repositories. The paper concludes with a discussion of the implications of DB-GPT framework on the future of human-database interaction and outlines potential avenues for further enhancements and applications in the field. The project code is available at https://github.com/eosphoros-ai/DB-GPT. Experience DB-GPT for yourself by installing it with the instructions https://github.com/eosphoros-ai/DB-GPT#install and view a concise 10-minute video at https://www.youtube.com/watch?v=KYs4nTDzEhk. | cs | 0 |
Title: Towards Abstract Wiener Model Spaces
Abstract: Wiener spaces are in many ways the decisive setting for fundamental results on Gaussian measures: large deviations (Schilder), quasi-invariance (Cameron--Martin), differential calculus (Malliavin), support description (Stroock--Varadhan), concentration of measure (Fernique), ... Analogues of these classical results have been derived in the "enhanced" context of Gaussian rough paths and, more recently, regularity structures equipped with Gaussian models. The aim of this article is to propose a notion of "abstract Wiener model space" that encompasses the aforementioned. More specifically, we focus here on enhanced Schilder type results, Cameron--Martin shifts and Fernique estimates, offering a somewhat unified view on results in Friz--Victoir 2007 and Hairer--Weber 2015. | math | 0 |
Title: Charatheodory and Smirnov type theorem for harmonic mappings
Abstract: We prove a version of Smirnov type theorem and Charatheodory type theorem for a harmonic homeomorphism of the unit disk onto a Jordan surface with rectifiable boundary. Further we establish the classical isoperimetric inequality and Riesz--Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptions of the boundary. | math | 1 |
Title: Minimizing the Weighted Number of Tardy Jobs is W[1]-hard
Abstract: We consider the $1||\sum w_J U_j$ problem, the problem of minimizing the weighted number of tardy jobs on a single machine. This problem is one of the most basic and fundamental problems in scheduling theory, with several different applications both in theory and practice. We prove that $1||\sum w_J U_j$ is W[1]-hard with respect to the number $p_{\#}$ of different processing times in the input, as well as with respect to the number $w_{\#}$ of different weights in the input. This, along with previous work, provides a complete picture for $1||\sum w_J U_j$ from the perspective of parameterized complexity, as well as almost tight complexity bounds for the problem under the Exponential Time Hypothesis (ETH). | cs | 0 |
Title: (Universal) Unconditional Verifiability in E-Voting without Trusted Parties
Abstract: In traditional e-voting protocols, privacy is often provided by a trusted authority that learns the votes and computes the tally. Some protocols replace the trusted authority by a set of authorities, and privacy is guaranteed if less than a threshold number of authorities are corrupt. For verifiability, stronger security guarantees are demanded. Typically, corrupt authorities that try to fake the result of the tally must always be detected. To provide verifiability, many e-voting protocols use Non-Interactive Zero-Knowledge proofs (NIZKs). Thanks to their non-interactive nature, NIZKs allow anybody, including third parties that do not participate in the protocol, to verify the correctness of the tally. Therefore, NIZKs can be used to obtain universal verifiability. Additionally, NIZKs also improve usability because they allow voters to cast a vote using a non-interactive protocol. The disadvantage of NIZKs is that their security is based on setup assumptions such as the common reference string (CRS) or the random oracle (RO) model. The former requires a trusted party for the generation of a common reference string. The latter, though a popular methodology for designing secure protocols, has been shown to be unsound. In this paper, we address the design of an e-voting protocol that provides verifiability without any trust assumptions, where verifiability here is meant without eligibility verification. We show that Non-Interactive Witness-Indistinguishable proofs (NIWI) can be used for this purpose. The e-voting scheme is private under the Decision Linear assumption, while verifiability holds unconditionally. To our knowledge, this is the first private e-voting scheme with perfect universal verifiability, i.e. one in which the probability of a fake tally not being detected is 0, and with {\em non-interactive} protocols that does not rely on trust assumptions. | cs | 1 |
Title: The anti-Ramsey numbers of cliques in complete multi-partite graphs
Abstract: A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Let $G$ and $H$ be two graphs. The anti-Ramsey number $\ar(G, H)$ is the maximum number of colors of an edge-coloring of $G$ that does not contain a rainbow copy of $H$. In this paper, we study the anti-Ramsey numbers of $K_k$ in complete multi-partite graphs. We determine the values of the anti-Ramsey numbers of $K_k$ in complete $k$-partite graphs and in balanced complete $r$-partite graphs for $r\geq k$. | math | 0 |
Title: Radical subgroups of finite reductive groups
Abstract: Radical subgroups play an important role in both group theory and representation theory. In this paper we present a strategy of classifying radical subgroups of finite reductive groups. As an application, we complete the proof of the inductive blockwise Alperin weight condition for the Chevalley groups $\F_4(q)$, contributing to the program to prove the Alperin weight conjecture by verifying its inductive condition for simple groups. | math | 0 |
Title: Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
Abstract: We prove that under n^{1/3} scaling, the limiting distribution as n goes to infinity of the free energy of Seppalainen's log-Gamma discrete directed polymer is GUE Tracy-Widom. The main technical innovation we provide is a general identity between a class of n-fold contour integrals and a class of Fredholm determinants. Applying this identity to the integral formula proved in [Corwin-O'Connell-Seppalainen-Zygouras] for the Laplace transform of the log-Gamma polymer partition function, we arrive at a Fredholm determinant which lends itself to asymptotic analysis (and thus yields the free energy limit theorem). The Fredholm determinant was anticipated in [Borodin-Corwin] via the formalism of Macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach. | math | 1 |
Title: Uniform regularity estimates and invisicid limit for the compressible non-resistive magnetohydrodynamics system
Abstract: We are concerned with the uniform regularity estimates of solutions to the two dimensional compressible non-resistive magnetohydrodynamics (MHD) equations with the no-slip boundary condition on velocity in the half plane. Under the assumption that the initial magnetic field is transverse to the boundary, the uniform conormal energy estimates are established for the solutions to compressible MHD equations with respect to small viscosity coefficients. As a direct consequence, we proved the inviscid limit of solutions from viscous MHD systems to the ideal MHD systems in $L^\infty$ sense. It shows that the transverse magnetic field can prevent the boundary layers from occurring in some physical regime. | math | 1 |
Title: On internal categories and crossed objects in the category of monoids
Abstract: It is a well-known fact that the category $\mathsf{Cat}(\mathbf{C})$ of internal categories in a category $\mathbf{C}$ has a description in terms of crossed modules, when $\mathbf{C}=\mathbf{Gr}$ is the category of groups. The proof of this result heavily uses the fact that any split epimorphism decomposes as a semi-direct product. An equivalent statement does not hold in the category $\mathbf{Mon}$ of monoids. In a previous work on quadratic algebras, I constructed an internal category in the category of monoids, see Section 6. Based on this construction, this paper will introduce the notion of a crossed semi-bimodule and show that it gives rise to an object in $\mathsf{Cat}(\mathbf{Mon})$. I will also relate this new notion to the crossed semi-modules introduced earlier by A. Patchkoria. | math | 0 |
Title: Filtered fiber functors over a general base
Abstract: We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor. | math | 0 |
Title: Rotor-routing reachability is easy, chip-firing reachability is hard
Abstract: Chip-firing and rotor-routing are two well-studied examples of abelian networks. We study the complexity of their respective reachability problems. We show that the rotor-routing reachability problem is decidable in polynomial time, and we give a simple characterization of when a chip-and-rotor configuration is reachable from another one. For chip-firing, it has been known that the reachability problem is in P if we have a class of graphs whose period length is polynomial (for example, Eulerian digraphs). Here we show that in the general case, chip-firing reachability is hard in the sense that if the chip-firing reachability problem were in P for general digraphs, then the polynomial hierarchy would collapse to NP. We encode graphs by their adjacency matrix, and we encode ribbon structures "succinctly", only remembering the number of consecutive parallel edges. | math | 1 |
Title: Correlations of the divisor function
Abstract: In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d). | math | 1 |
Title: Matchings in hypercubes extend to long cycles
Abstract: The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be extended to a Hamilton cycle, i.e., to a cycle that visits every vertex exactly once. We prove that every matching of $Q_d$, $d\ge 2$, can be extended to a cycle that visits at least a $2/3$-fraction of all vertices. | math | 0 |
Title: Vectorization of Multibyte Floating Point Data Formats
Abstract: We propose a scheme for reduced-precision representation of floating point data on a continuum between IEEE-754 floating point types. Our scheme enables the use of lower precision formats for a reduction in storage space requirements and data transfer volume. We describe how our scheme can be accelerated using existing hardware vector units on a general-purpose processor (GPP). Exploiting native vector hardware allows us to support reduced precision floating point with low overhead. We demonstrate that supporting reduced precision in the compiler as opposed to using a library approach can yield a low overhead solution for GPPs. | cs | 1 |
Title: Sampling projections in the uniform norm
Abstract: We show that there are sampling projections on arbitrary $n$-dimensional subspaces of $B(D)$ with at most $2n$ samples and norm of order $\sqrt{n}$, where $B(D)$ is the space of complex-valued bounded functions on a set $D$. This gives a more explicit form of the Kadets-Snobar theorem for the uniform norm and improves upon Auerbach's lemma. We discuss consequences for optimal recovery in $L_p$. | math | 0 |
Title: Near Real-Time Data-Driven Control of Virtual Reality Traffic in Open Radio Access Network
Abstract: In mobile networks, Open Radio Access Network (ORAN) provides a framework for implementing network slicing that interacts with the resources at the lower layers. Both monitoring and Radio Access Network (RAN) control is feasible for both 4G and 5G systems. In this work, we consider how data-driven resource allocation in a 4G context can enable adaptive slice allocation to steer the experienced latency of Virtual Reality (VR) traffic towards a requested latency. We develop an xApp for the near real-time RAN Intelligent Controller (RIC) that embeds a heuristic algorithm for latency control, aiming to: (1) maintain latency of a VR stream around a requested value; and (2) improve the available RAN allocation to offer higher bit rate to another user. We have experimentally demonstrated the proposed approach in an ORAN testbed. Our results show that the data-driven approach can dynamically follow the variation of the traffic load while satisfying the required latency. This results in 15.8% more resources to secondary users than a latency-equivalent static allocation. | cs | 0 |
Title: Reconstruction of curves from their theta hyperplanes in genera $6$ and $7$
Abstract: We derive a formula for reconstructing a generic complex canonical curve $C$ of genus 6 and 7 in terms of the theta hyperplanes of $C$. Hence, we get a generic inverse to the Torelli map, as well as a complete description of the Schottky locus in these genera. The computational part of the proof relies on a certified numerical argument. | math | 0 |
Title: The Evolving Ecosystem of Predatory Journals: A Case Study in Indian Perspective
Abstract: Digital advancement in scholarly repositories has led to the emergence of a large number of open access predatory publishers that charge high article processing fees from authors but fail to provide necessary editorial and publishing services. Identifying and blacklisting such publishers has remained a research challenge due to the highly volatile scholarly publishing ecosystem. This paper presents a data-driven approach to study how potential predatory publishers are evolving and bypassing several regularity constraints. We empirically show the close resemblance of predatory publishers against reputed publishing groups. In addition to verifying standard constraints, we also propose distinctive signals gathered from network-centric properties to understand this evolving ecosystem better. | cs | 1 |
Title: Quantization effects for multi-component Ginzburg-Landau vortices
Abstract: In this paper, we are concerned with $n$-component Ginzburg-Landau equations on $\rtwo$.By introducing a diffusion constant for each component, we discuss that the $n$-component equations are different from $n$-copies of the single Ginzburg-Landau equations.Then, the results of Brezis-Merle-Riviere for the single Ginzburg-Landau equation can be nontrivially extended to the multi-component case.First, we show that if the solutions have their gradients in $L^2$ space, they are trivial solutions.Second, we prove that if the potential is square summable, then it has quantized integrals, i.e., there exists one-to-one correspondence between the possible values of the potential energy and $\nat^n$.Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of $n$-component equations. | math | 0 |
Title: Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs
Abstract: A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity $n^{O(\log n)}$. In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-$3$ circuits. The prior results for the sum of constantly many set-multilinear depth-$3$ circuits were only slightly better than brute-force, i.e. exponential-time. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives). | cs | 1 |
Title: Post-hoc evaluation of nodes influence in information cascades: the case of coordinated accounts
Abstract: In the last years, social media has gained an unprecedented amount of attention, playing a pivotal role in shaping the contemporary landscape of communication and connection. However, Coordinated Inhautentic Behaviour (CIB), defined as orchestrated efforts by entities to deceive or mislead users about their identity and intentions, has emerged as a tactic to exploit the online discourse. In this study, we quantify the efficacy of CIB tactics by defining a general framework for evaluating the influence of a subset of nodes in a directed tree. We design two algorithms that provide optimal and greedy post-hoc placement strategies that lead to maximising the configuration influence. We then consider cascades from information spreading on Twitter to compare the observed behaviour with our algorithms. The results show that, according to our model, coordinated accounts are quite inefficient in terms of their network influence, thus suggesting that they may play a less pivotal role than expected. Moreover, the causes of these poor results may be found in two separate aspects: a bad placement strategy and a scarcity of resources. | cs | 0 |
Title: The Cauchy problem on large time for the Water Waves equations with large topography variations
Abstract: We prove the local existence for the Water Waves equations with large bathymetric variations on a time interval of size 1/\epsilon, where $\epsilon$ measures the amplitude of the wave. We just need the presence of surface tension. | math | 1 |
Title: Superimposed Pilots are Superior for Mitigating Pilot Contamination in Massive MIMO
Abstract: In this paper, superimposed pilots are introduced as an alternative to time-multiplexed pilot and data symbols for mitigating pilot contamination in massive multiple-input multiple-output (MIMO) systems. We propose a non-iterative scheme for uplink channel estimation based on superimposed pilots and derive an expression for the uplink signal-to-interference-plus-noise ratio (SINR) at the output of a matched filter employing this channel estimate. Based on this expression, we observe that power control is essential when superimposed pilots are employed. Moreover, the quality of the channel estimate can be improved by reducing the interference that results from transmitting data alongside the pilots, and an intuitive iterative data-aided scheme that reduces this component of interference is also proposed. Approximate expressions for the uplink SINR are provided for the iterative data-aided method as well. In addition, we show that a hybrid system with users utilizing both time-multiplexed and superimposed pilots is superior to an optimally designed system that employs only time-multiplexed pilots, even when the non-iterative channel estimate is used to build the detector and precoder. We also describe a simple approach to implement this hybrid system by minimizing the overall inter and intra-cell interference. Numerical simulations demonstrating the performance of the proposed channel estimation schemes and the superiority of the hybrid system are also provided. | cs | 1 |
Title: Global existence and Hadamard differentiability of hysteresis-reaction-diffusion systems
Abstract: We consider a class of semilinear parabolic evolution equations subject to a hysteresis operator and a Bochner-Lebesgue integrable source term. The underlying spatial domain is allowed to have a very general boundary. In the first part of the paper, we apply semigroup theory to prove well-posedness and boundedness of the solution operator. Rate independence in reaction-diffusion systems complicates the analysis, since the reaction term acts no longer local in time. This demands careful estimates when working with semigroup methods. In the second part, we show Lipschitz continuity and Hadamard differentiability of the solution operator. We use fixed point arguments to derive a representation for the derivative in terms of the evolution system. Finally, we apply our results to an optimal control problem in which the source term acts as a control function and show existence of an optimal solution. | math | 1 |
Title: Outage Analysis for Active Reconfigurable Intelligent Surface-Enhanced Wireless Powered Communication Networks
Abstract: Wireless powered communication (WPC) involves the integration of energy harvesting and data transmission. This allows devices to communicate without constant battery replacements or wired power sources. Reconfigurable intelligent surfaces (RISs) can dynamically manipulate radio signals. In this paper, we explore the use of active elements to mitigate double-fading challenges inherent in RIS-aided links. We enhance the reliability performance for an energy-constrained user by combining active RIS and WPC. The theoretical closed-form analysis, which includes transmission rate, harvested energy, and outage probability, provides valuable insights that inform parameter selection. | cs | 0 |
Title: Classification and Treatment Learning with Constraints via Composite Heaviside Optimization: a Progressive MIP Method
Abstract: This paper proposes a Heaviside composite optimization approach and presents a progressive (mixed) integer programming (PIP) method for solving multi-class classification and multi-action treatment problems with constraints. A Heaviside composite function is a composite of a Heaviside function (i.e., the indicator function of either the open $( \, 0,\infty )$ or closed $[ \, 0,\infty \, )$ interval) with a possibly nondifferentiable function. Modeling-wise, we show how Heaviside composite optimization provides a unified formulation for learning the optimal multi-class classification and multi-action treatment rules, subject to rule-dependent constraints stipulating a variety of domain restrictions. A Heaviside composite function has an equivalent discrete formulation %in terms of integer variables, and the resulting optimization problem can in principle be solved by integer programming (IP) methods. Nevertheless, for constrained learning problems with large data sets, %of modest or large sizes, a straightforward application of off-the-shelf IP solvers is usually ineffective in achieving global optimality. To alleviate such a computational burden, our major contribution is the proposal of the PIP method by leveraging the effectiveness of state-of-the-art IP solvers for problems of modest sizes. We provide the theoretical advantage of the PIP method with the connection to continuous optimization and show that the computed solution is locally optimal for a broad class of Heaviside composite optimization problems. The numerical performance of the PIP method is demonstrated by extensive computational experimentation. | math | 0 |
Title: LLM Harmony: Multi-Agent Communication for Problem Solving
Abstract: Large Language Models (LLMs) have revolutionized Natural Language Processing but exhibit limitations, particularly in autonomously addressing novel challenges such as reasoning and problem-solving. Traditional techniques like chain-of-thought prompting necessitate explicit human guidance. This paper introduces a novel multi-agent communication framework, inspired by the CAMEL model, to enhance LLMs' autonomous problem-solving capabilities. The framework employs multiple LLM agents, each with a distinct persona, engaged in role-playing communication, offering a nuanced and adaptable approach to diverse problem scenarios. Extensive experimentation demonstrates the framework's superior performance and adaptability, providing valuable insights into the collaborative potential of multiple agents in overcoming the limitations of individual models. | cs | 0 |
Title: Beyond Efficiency: A Systematic Survey of Resource-Efficient Large Language Models
Abstract: The burgeoning field of Large Language Models (LLMs), exemplified by sophisticated models like OpenAI's ChatGPT, represents a significant advancement in artificial intelligence. These models, however, bring forth substantial challenges in the high consumption of computational, memory, energy, and financial resources, especially in environments with limited resource capabilities. This survey aims to systematically address these challenges by reviewing a broad spectrum of techniques designed to enhance the resource efficiency of LLMs. We categorize methods based on their optimization focus: computational, memory, energy, financial, and network resources and their applicability across various stages of an LLM's lifecycle, including architecture design, pretraining, finetuning, and system design. Additionally, the survey introduces a nuanced categorization of resource efficiency techniques by their specific resource types, which uncovers the intricate relationships and mappings between various resources and corresponding optimization techniques. A standardized set of evaluation metrics and datasets is also presented to facilitate consistent and fair comparisons across different models and techniques. By offering a comprehensive overview of the current sota and identifying open research avenues, this survey serves as a foundational reference for researchers and practitioners, aiding them in developing more sustainable and efficient LLMs in a rapidly evolving landscape. | cs | 0 |
Title: Enhancing NOMA Networks via Reconfigurable Multi-Functional Surface
Abstract: By flexibly manipulating the radio propagation environment, reconfigurable intelligent surface (RIS) is a promising technique for future wireless communications. However, the single-side coverage and double-fading attenuation faced by conventional RISs largely restrict their applications. To address this issue, we propose a novel concept of multi-functional RIS (MF-RIS), which provides reflection, transmission, and amplification simultaneously for the incident signal. With the aim of enhancing the performance of a non-orthogonal multiple-access (NOMA) downlink multiuser network, we deploy an MF-RIS to maximize the sum rate by jointly optimizing the active beamforming and MF-RIS coefficients. Then, an alternating optimization algorithm is proposed to solve the formulated non-convex problem by exploiting successive convex approximation and penalty-based method. Numerical results show that the proposed MF-RIS outperforms conventional RISs under different settings. | cs | 1 |
Title: Fast Certification of Vision-Language Models Using Incremental Randomized Smoothing
Abstract: A key benefit of deep vision-language models such as CLIP is that they enable zero-shot open vocabulary classification; the user has the ability to define novel class labels via natural language prompts at inference time. However, while CLIP-based zero-shot classifiers have demonstrated competitive performance across a range of domain shifts, they remain highly vulnerable to adversarial attacks. Therefore, ensuring the robustness of such models is crucial for their reliable deployment in the wild. In this work, we introduce Open Vocabulary Certification (OVC), a fast certification method designed for open-vocabulary models like CLIP via randomized smoothing techniques. Given a base "training" set of prompts and their corresponding certified CLIP classifiers, OVC relies on the observation that a classifier with a novel prompt can be viewed as a perturbed version of nearby classifiers in the base training set. Therefore, OVC can rapidly certify the novel classifier using a variation of incremental randomized smoothing. By using a caching trick, we achieve approximately two orders of magnitude acceleration in the certification process for novel prompts. To achieve further (heuristic) speedups, OVC approximates the embedding space at a given input using a multivariate normal distribution bypassing the need for sampling via forward passes through the vision backbone. We demonstrate the effectiveness of OVC on through experimental evaluation using multiple vision-language backbones on the CIFAR-10 and ImageNet test datasets. | cs | 0 |
Title: Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity
Abstract: We address the persistence of regularity for the 2D $\alpha$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_{0}, \rho_{0}) \in W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$, where $s> 1$ and $q \in (2, \infty)$. We prove that the solution $(u(t), \rho(t))$ exists and belongs to $W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$ for all positive time $t$ for $q>2$, where $\alpha\in(1,2)$ is arbitrary. | math | 1 |
Title: The specialization index of a variety over a discretely valued field
Abstract: Let $X$ be a proper variety over a henselian discretely valued field. An important obstruction to the existence of a rational point on $X$ is the index, the minimal positive degree of a zero cycle on $X$. This paper introduces a new invariant, the specialization index, which is a closer approximation of the existence of a rational point. We provide an explicit formula for the specialization index in terms of an $snc$-model, and we give examples of curves where the index equals one but the specialization index is different from one, and thus explains the absence of a rational point. Our main result states that the specialization index of a smooth, proper, geometrically connected $\mathbb{C}((t))$-variety with trivial coherent cohomology is equal to one. | math | 1 |
Title: When ideals properly extend the class of Arbault sets
Abstract: In this article we continue the investigation of generalized version of Arbault sets, that was initiated in \cite{DGT} but look at the picture from the most general point of view where ideals come into play. While Arbault sets can be naturally associated with the Frechet ideal $Fin$, in \cite{DGT} it was observed that when $Fin$ is replaced by the natural density ideal $\mathcal{I}_d$ one can obtain a strictly larger class of trigonometric thin sets containing Arbault sets. From the set theoretic point of view a natural question arises as whether one can broaden the picture and specify a class of ideals (instead of a single ideal) each of which would have the similar effect. As a natural candidate, we focus on a special class of ideals, namely, non-$snt$ ideals ($snt$ stands for ``strongly non translation invariant") which properly contains the class of translation invariant ideals ($\varsupsetneq Fin$) and happens to contain ideals generated by simple density functions as also certain non-negative regular summability matrices (but not all) which can be seen from \cite{DG6}. We consider the resulting class of $\mathcal{I}$-Arbault sets and it is observed that for each such ideal, the class of $\mathcal{I}$-Arbault sets not only properly contains the class of classical Arbault sets \cite{Ar} but also a large subfamily of $\mathbf{N}$-sets (also called ``sets of absolute convergence") \cite{Ft} while being contained in the class of weak Dirichlet sets. %In particular it properly contains the family of $\mathbf{N}_0$-sets which have been extensively used in the literature (see \cite{Ar, Ka, Ko}). Though distinct from the class of $\mathbf{N}$-sets, this happens to be a new class strictly lying between the class of Arbault sets and the class of weak Dirichlet sets. | math | 0 |
Title: Two trees are better than one
Abstract: We consider partitions of a point set into two parts, and the lengths of the minimum spanning trees of the original set and of the two parts. If $w(P)$ denotes the length of a minimum spanning tree of $P$, we show that every set $P$ of $n \geq 12$ points admits a bipartition $P= R \cup B$ for which the ratio $\frac{w(R)+w(B)}{w(P)}$ is strictly larger than $1$; and that $1$ is the largest number with this property. Furthermore, we provide a very fast algorithm that computes such a bipartition in $O(1)$ time and one that computes the corresponding ratio in $O(n \log{n})$ time. In certain settings, a ratio larger than $1$ can be expected and sometimes guaranteed. For example, if $P$ is a set of $n$ random points uniformly distributed in $[0,1]^2$ ($n \to \infty$), then for any $\eps>0$, the above ratio in a maximizing partition is at least $\sqrt2 -\eps$ with probability tending to $1$. As another example, if $P$ is a set of $n$ points with spread at most $\alpha \sqrt{n}$, for some constant $\alpha>0$, then the aforementioned ratio in a maximizing partition is $1 + \Omega(\alpha^{-2})$. All our results and techniques are extendable to higher dimensions. | cs | 0 |
Title: Existence of solutions to the nonlinear equations characterizing the precise error of M-estimators
Abstract: Major progress has been made in the previous decade to characterize the asymptotic behavior of regularized M-estimators in high-dimensional regression problems in the proportional asymptotic regime where the sample size $n$ and the number of features $p$ are increasing simultaneously such that $n/p\to \delta \in(0,\infty)$, using powerful tools such as Approximate Message Passing or the Convex Gaussian Min-Max Theorem (CGMT). The asymptotic error and behavior of the regularized M-estimator is then typically described by a system of nonlinear equations with a few scalar unknowns, and the solution to this system precisely characterize the asymptotic error. Application of the CGMT and related machinery requires the existence of a solution to this low-dimensional system of equations. This paper resolves the question of existence of solution to this low-dimensional system for the case of linear models with independent additive noise, when both the data-fitting loss function and regularization penalty are separable and convex. Such existence result for solution to the nonlinear system were previously known under strong convexity for specific estimators such as the Lasso. The main idea behind this existence result is inspired by an argument developed \cite{montanari2019generalization,celentano2020lasso} in different contexts: By constructing an ad-hoc convex minimization problem in an infinite dimensional Hilbert space, the existence of the Lagrange multiplier for this optimization problem makes it possible to construct explicitly solutions to the low-dimensional system of interest. The conditions under which we derive this existence result exactly correspond to the side of the phase transition where perfect recovery $\hat x= x_0$ fails, so that these conditions are optimal. | math | 0 |
Title: Maximal Sections of Sheaves of Data over an Abstract Simplicial Complex
Abstract: We employ techniques from topological data analysis to model sensor networks. Our approach to sensor integration uses the topological method of sheaves over cell complexes. The internal consistency of data from individual sensors is determined by a set of consistency functions assigned to elements of the complex. Using these functions we determine, for any collection of data, the unique set of maximal sections of consistent data received from the sensors. We offer a proof for the existence and uniqueness of these sections and illustrate the ideas with examples. | cs | 1 |
Title: Boolean TQFTs with accumulating defects, sofic systems, and automata for infinite words
Abstract: Any finite state automaton gives rise to a Boolean one-dimensional TQFT with defects and inner endpoints of cobordisms. This paper extends the correspondence to Boolean TQFTs where defects accumulate toward inner endpoints, relating such TQFTs and topological theories to sofic systems and $\omega$-automata. | math | 0 |
Title: Calculus and applications
Abstract: This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and of the integral $\int_a^bf(x)dx$. Then we investigate the case of the complex functions $f:\mathbb C\to\mathbb C$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:\mathbb R^N\to\mathbb R^M$ or $f:\mathbb R^N\to\mathbb C^M$ or $f:\mathbb C^N\to\mathbb C^M$, with general theory, integration results, maximization questions, and basic applications to physics. | math | 0 |
Title: Explicit Generators for the Stabilizers of Rational Points in Thompson's Group $F$
Abstract: We construct explicit finite generating sets for the stabilizers in Thompson's group $F$ of rational points of a unit interval or a Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in $F$. It is well known that the stabilizers of dyadic rational points are isomorphic to $F\times F$ and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point $b\in (0,1)$ is generated by 5 elements that are explicitly calculated as words in generators $x_0, x_1$ of $F$ that depend on the binary expansion of $b$. We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented. | math | 0 |
Title: Lorentzian connections with parallel twistor-free torsion
Abstract: We describe Lorentzian manifolds that admit metric connections with parallel torsion having zero twistorial component and non-zero vectorial component. We also describe Lorentzian manifolds admitting metric connections with closed parallel skew-symmetric torsion. | math | 0 |
Title: The convolution algebra of an absolutely locally compact topos
Abstract: We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra $\mathcal{C}_c(\mathcal{T},A)$ which is well defined up to Morita equivalence and characterized by the fact that the category of non-degenerate modules over $\mathcal{C}_c(\mathcal{T},A)$ is equivalent to the category of sheaf of $A$-module over $\mathcal{T}$. In the case where $A$ is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive algebra that allows to complete it in various Banach and $C^*$-algebras: $L^1(\mathcal{T},A)$, $C^*_{red}(\mathcal{T},A)$ and $C^*_{max}(\mathcal{T},A)$. We also give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras. | math | 1 |
Title: Quantifying Deep Learning Model Uncertainty in Conformal Prediction
Abstract: Precise estimation of predictive uncertainty in deep neural networks is a critical requirement for reliable decision-making in machine learning and statistical modeling, particularly in the context of medical AI. Conformal Prediction (CP) has emerged as a promising framework for representing the model uncertainty by providing well-calibrated confidence levels for individual predictions. However, the quantification of model uncertainty in conformal prediction remains an active research area, yet to be fully addressed. In this paper, we explore state-of-the-art CP methodologies and their theoretical foundations. We propose a probabilistic approach in quantifying the model uncertainty derived from the produced prediction sets in conformal prediction and provide certified boundaries for the computed uncertainty. By doing so, we allow model uncertainty measured by CP to be compared by other uncertainty quantification methods such as Bayesian (e.g., MC-Dropout and DeepEnsemble) and Evidential approaches. | cs | 0 |
Title: Person Re-identification: Implicitly Defining the Receptive Fields of Deep Learning Classification Frameworks
Abstract: The \emph{receptive fields} of deep learning classification models determine the regions of the input data that have the most significance for providing correct decisions. The primary way to learn such receptive fields is to train the models upon masked data, which helps the networks to ignore any unwanted regions, but has two major drawbacks: 1) it often yields edge-sensitive decision processes; and 2) augments the computational cost of the inference phase considerably. This paper describes a solution for implicitly driving the inference of the networks' receptive fields, by creating synthetic learning data composed of interchanged segments that should be \emph{apriori} important/irrelevant for the network decision. In practice, we use a segmentation module to distinguish between the foreground (important)/background (irrelevant) parts of each learning instance, and randomly swap segments between image pairs, while keeping the class label exclusively consistent with the label of the deemed important segments. This strategy typically drives the networks to early convergence and appropriate solutions, where the identity and clutter descriptions are not correlated. Moreover, this data augmentation solution has various interesting properties: 1) it is parameter-free; 2) it fully preserves the label information; and, 3) it is compatible with the typical data augmentation techniques. In the empirical validation, we considered the person re-identification problem and evaluated the effectiveness of the proposed solution in the well-known \emph{Richly Annotated Pedestrian} (RAP) dataset for two different settings (\emph{upper-body} and \emph{full-body}), observing highly competitive results over the state-of-the-art. Under a reproducible research paradigm, both the code and the empirical evaluation protocol are available at \url{https://github.com/Ehsan-Yaghoubi/reid-strong-baseline}. | cs | 1 |
Title: Acyclicity of Preferences, Nash Equilibria, and Subgame Perfect Equilibria: a Formal and Constructive Equivalence
Abstract: In 1953, Kuhn showed that every sequential game has a Nash equilibrium by showing that a procedure, named ``backward induction'' in game theory, yields a Nash equilibrium. It actually yields Nash equilibria that define a proper subclass of Nash equilibria. In 1965, Selten named this proper subclass subgame perfect equilibria. In game theory, payoffs are rewards usually granted at the end of a game. Although traditional game theory mainly focuses on real-valued payoffs that are implicitly ordered by the usual total order over the reals, works of Simon or Blackwell already involved partially ordered payoffs. This paper generalises the notion of sequential game by replacing real-valued payoff functions with abstract atomic objects, called outcomes, and by replacing the usual total order over the reals with arbitrary binary relations over outcomes, called preferences. This introduces a general abstract formalism where Nash equilibrium, subgame perfect equilibrium, and ``backward induction'' can still be defined. This paper proves that the following three propositions are equivalent: 1) Preferences over the outcomes are acyclic. 2) Every sequential game has a Nash equilibrium. 3) Every sequential game has a subgame perfect equilibrium. The result is fully computer-certified using Coq. Beside the additional guarantee of correctness, the activity of formalisation using Coq also helps clearly identify the useful definitions and the main articulations of the proof. | cs | 1 |
Title: ClassWise-SAM-Adapter: Parameter Efficient Fine-tuning Adapts Segment Anything to SAR Domain for Semantic Segmentation
Abstract: In the realm of artificial intelligence, the emergence of foundation models, backed by high computing capabilities and extensive data, has been revolutionary. Segment Anything Model (SAM), built on the Vision Transformer (ViT) model with millions of parameters and vast training dataset SA-1B, excels in various segmentation scenarios relying on its significance of semantic information and generalization ability. Such achievement of visual foundation model stimulates continuous researches on specific downstream tasks in computer vision. The ClassWise-SAM-Adapter (CWSAM) is designed to adapt the high-performing SAM for landcover classification on space-borne Synthetic Aperture Radar (SAR) images. The proposed CWSAM freezes most of SAM's parameters and incorporates lightweight adapters for parameter efficient fine-tuning, and a classwise mask decoder is designed to achieve semantic segmentation task. This adapt-tuning method allows for efficient landcover classification of SAR images, balancing the accuracy with computational demand. In addition, the task specific input module injects low frequency information of SAR images by MLP-based layers to improve the model performance. Compared to conventional state-of-the-art semantic segmentation algorithms by extensive experiments, CWSAM showcases enhanced performance with fewer computing resources, highlighting the potential of leveraging foundational models like SAM for specific downstream tasks in the SAR domain. The source code is available at: https://github.com/xypu98/CWSAM. | cs | 0 |
Title: Almost spanning distance trees in subsets of finite vector spaces
Abstract: For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined as $\sum_{i=1}^d (x_i-y_i)^2$. A distance graph is a graph associated with a non-zero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every nearly spanning distance tree with bounded degree in each distance. This quantitatively improves results by Bennett, Chapman, Covert, Hart, Iosevich, and Pakianathan on finding distance paths, and results by Pham, Senger, Tait, and Thu on finding distance trees. A key ingredient in proving our main result is to obtain a colorful generalization of a classical result of Haxell about finding nearly spanning bounded-degree trees in an expander. | math | 0 |
Title: Extremal spectral radius of nonregular graphs with prescribed maximum degree
Abstract: Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $\Delta$. Let $\lambda_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the largest eigenvalue of non-regular graphs, J. Combin. Theory Ser. B, 97 (2007) 1010--1018] asserts that \[ \lim_{n\to\infty} \frac{n^2(\Delta-\lambda_1(G))}{\Delta-1} = \pi^2 \] for each fixed $\Delta$. Concerning an important structural property of the extremal graphs $G$, Liu and Li present another conjecture which states that $G$ has degree sequence $\Delta,\ldots,\Delta,\delta$. Here, $\delta=\Delta-1$ or $\delta=\Delta-2$ depending on the parity of $n\Delta$. In this paper, we make progress on the two conjectures. To be precise, we disprove the first conjecture for all $\Delta\geq 3$ by showing that the limit superior is at most $\pi^2/2$. For small $\Delta$, we determine the precise asymptotic behavior of $\Delta-\lambda_1(G)$. In particular, we show that $\lim\limits_{n\to\infty} n^2 (\Delta - \lambda_1(G)) /(\Delta - 1) = \pi^2/4$ if $\Delta=3$; and $\lim\limits_{n\to\infty} n^2 (\Delta - \lambda_1(G)) /(\Delta - 2) = \pi^2/2$ if $\Delta = 4$. We also confirm the second conjecture for $\Delta = 3$ and $\Delta = 4$ by determining the precise structure of extremal graphs. Particularly, we show that the extremal graphs for $\Delta\in\{3,4\}$ must have a path-like structure built from specific blocks. | math | 1 |
Title: Explicit stabilized multirate methods for the monodomain model in cardiac electrophysiology
Abstract: Fully explicit stabilized multirate (mRKC) methods are well-suited for the numerical solution of large multiscale systems of stiff ordinary differential equations thanks to their improved stability properties. To demonstrate their efficiency for the numerical solution of stiff, multiscale, nonlinear parabolic PDE's, we apply mRKC methods to the monodomain equation from cardiac electrophysiology. In doing so, we propose an improved version, specifically tailored to the monodomain model, which leads to the explicit exponential multirate stabilized (emRKC) method. Several numerical experiments are conducted to evaluate the efficiency of both mRKC and emRKC, while taking into account different finite element meshes (structured and unstructured) and realistic ionic models. The new emRKC method typically outperforms a standard implicit-explicit baseline method for cardiac electrophysiology. Code profiling and strong scalability results further demonstrate that emRKC is faster and inherently parallel without sacrificing accuracy. | math | 0 |
Title: A stochastic approximation scheme and convergence theorem for particle interactions with perfectly reflecting boundaries
Abstract: We prove the existence of a solution to an equation governing the number density within a compact domain of a discrete particle system for a prescribed class of particle interactions taking into account the effects of the diffusion and drift of the set of particles. Each particle carries a number of internal coordinates which may evolve continuously in time, determined by what we will refer to as the internal drift, or discretely via the interaction kernels. Perfectly reflecting boundary conditions are imposed on the system and all the processes may be spatially and temporally inhomogeneous. We use a relative compactness argument to construct a sequence of measures that converge weakly to a solution of the governing equation. Since the proof of existence is a constructive one, it provides a stochastic approximation scheme that can be used for the numerical study of molecular dynamics. | math | 1 |
Title: The Implicit Bias of Benign Overfitting
Abstract: The phenomenon of benign overfitting, where a predictor perfectly fits noisy training data while attaining near-optimal expected loss, has received much attention in recent years, but still remains not fully understood beyond well-specified linear regression setups. In this paper, we provide several new results on when one can or cannot expect benign overfitting to occur, for both regression and classification tasks. We consider a prototypical and rather generic data model for benign overfitting of linear predictors, where an arbitrary input distribution of some fixed dimension $k$ is concatenated with a high-dimensional distribution. For linear regression which is not necessarily well-specified, we show that the minimum-norm interpolating predictor (that standard training methods converge to) is biased towards an inconsistent solution in general, hence benign overfitting will generally not occur. Moreover, we show how this can be extended beyond standard linear regression, by an argument proving how the existence of benign overfitting on some regression problems precludes its existence on other regression problems. We then turn to classification problems, and show that the situation there is much more favorable. Specifically, we prove that the max-margin predictor (to which standard training methods are known to converge in direction) is asymptotically biased towards minimizing a weighted \emph{squared hinge loss}. This allows us to reduce the question of benign overfitting in classification to the simpler question of whether this loss is a good surrogate for the misclassification error, and use it to show benign overfitting in some new settings. | cs | 1 |
Title: Weierstrass Bridges
Abstract: We introduce a new class of stochastic processes called fractional Wiener-Weierstrass bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstrass functions to an underlying fractional Brownian bridge. By analyzing the $p$-th variation of the fractional Wiener-Weierstrass bridge along the sequence of $b$-adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener-Weierstrass bridges that are based on standard Brownian bridge. We furthermore prove that fractional Wiener-Weierstrass bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstrass bridges based on bridges derived from a general continuous Gaussian martingale. | math | 0 |
Title: Positroids, knots, and $q,t$-Catalan numbers
Abstract: We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over $\mathbb{F}_q$) to Khovanov--Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $\mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules. | math | 1 |
Title: Fredholm-type Operators and Index
Abstract: While in \cite{HB} we studied classes of Fredholm-type operators defined by the homomorphism $\Pi$ from $L(X)$ onto the Calkin algebra $\mathcal{C}(X)$, $X$ being a Banach space, we study in this paper two classes of Fredholm-type operators defined by the homomorphism $\pi$ from $L(X)$ onto the algebra $\mathcal{C}_0(X)= L(X)/F_0(X),$ where $F_0(X)$ is the ideal of finite rank operators in $L(X).$ Then we define an index for Fredholm-type operators and we show that this new index satisfies similar properties as the usual Fredholm index. | math | 0 |
Title: Unique Triangulated 1-Planar Graphs
Abstract: It is well-known that every 3-connected planar graph has a unique planar embedding on the sphere. We study the extension to triangulated 1-planar graphs, T1P graphs for short, which admit an embedding in which each edge is crossed at most once and each face is a triangle, and obtain an algorithmic solution by a cubic time recognition algorithm that also counts the number of T1P embeddings. In particular, we show that every triangulated planar graph has a unique T1P embedding, although it may admit many 1-planar embeddings, and that any 6-connected T1P graph has a unique 1-planar embedding, except for full generalized two-stars that admit two or eight 1-planar embeddings. Our algorithm extends, refines, and corrects a previous recognition algorithm by Chen, Grigni and Papadimitiou (``Recognizing Hole-Free 4-Map Graphs in Cubic Time'', Algorithmica 45 (2006)). | cs | 0 |
Title: On games and simulators as a platform for development of artificial intelligence for command and control
Abstract: Games and simulators can be a valuable platform to execute complex multi-agent, multiplayer, imperfect information scenarios with significant parallels to military applications: multiple participants manage resources and make decisions that command assets to secure specific areas of a map or neutralize opposing forces. These characteristics have attracted the artificial intelligence (AI) community by supporting development of algorithms with complex benchmarks and the capability to rapidly iterate over new ideas. The success of artificial intelligence algorithms in real-time strategy games such as StarCraft II have also attracted the attention of the military research community aiming to explore similar techniques in military counterpart scenarios. Aiming to bridge the connection between games and military applications, this work discusses past and current efforts on how games and simulators, together with the artificial intelligence algorithms, have been adapted to simulate certain aspects of military missions and how they might impact the future battlefield. This paper also investigates how advances in virtual reality and visual augmentation systems open new possibilities in human interfaces with gaming platforms and their military parallels. | cs | 1 |
Title: Sample Complexity Bounds for Two Timescale Value-based Reinforcement Learning Algorithms
Abstract: Two timescale stochastic approximation (SA) has been widely used in value-based reinforcement learning algorithms. In the policy evaluation setting, it can model the linear and nonlinear temporal difference learning with gradient correction (TDC) algorithms as linear SA and nonlinear SA, respectively. In the policy optimization setting, two timescale nonlinear SA can also model the greedy gradient-Q (Greedy-GQ) algorithm. In previous studies, the non-asymptotic analysis of linear TDC and Greedy-GQ has been studied in the Markovian setting, with diminishing or accuracy-dependent stepsize. For the nonlinear TDC algorithm, only the asymptotic convergence has been established. In this paper, we study the non-asymptotic convergence rate of two timescale linear and nonlinear TDC and Greedy-GQ under Markovian sampling and with accuracy-independent constant stepsize. For linear TDC, we provide a novel non-asymptotic analysis and show that it attains an $\epsilon$-accurate solution with the optimal sample complexity of $\mathcal{O}(\epsilon^{-1}\log(1/\epsilon))$ under a constant stepsize. For nonlinear TDC and Greedy-GQ, we show that both algorithms attain $\epsilon$-accurate stationary solution with sample complexity $\mathcal{O}(\epsilon^{-2})$. It is the first non-asymptotic convergence result established for nonlinear TDC under Markovian sampling and our result for Greedy-GQ outperforms the previous result orderwisely by a factor of $\mathcal{O}(\epsilon^{-1}\log(1/\epsilon))$. | cs | 1 |
Title: Periodic Strategies II: Generalizations and Extensions
Abstract: At a mixed Nash equilibrium, the payoff of a player does not depend on her own action, as long as her opponent sticks to his. In a periodic strategy, a concept developed in a previous paper (arXiv:1307.2035v4), in contrast, the own payoff does not depend on the opponent's action. Here, we generalize this to multi-player simultaneous perfect information strategic form games. We show that also in this class of games, there always exists at least one periodic strategy, and we investigate the mathematical properties of such periodic strategies. In addition, we demonstrate that periodic strategies may exist in games with incomplete information; we shall focus on Bayesian games. Moreover we discuss the differences between the periodic strategies formalism and cooperative game theory. In fact, the periodic strategies are obtained in a purely non-cooperative way, and periodic strategies are as cooperative as the Nash equilibria are. Finally, we incorporate the periodic strategies in an epistemic game theory framework, and discuss several features of this approach. | cs | 1 |
Title: Stochastic Approximation Approaches to Group Distributionally Robust Optimization
Abstract: This paper investigates group distributionally robust optimization (GDRO), with the purpose to learn a model that performs well over $m$ different distributions. First, we formulate GDRO as a stochastic convex-concave saddle-point problem, and demonstrate that stochastic mirror descent (SMD), using $m$ samples in each iteration, achieves an $O(m (\log m)/\epsilon^2)$ sample complexity for finding an $\epsilon$-optimal solution, which matches the $\Omega(m/\epsilon^2)$ lower bound up to a logarithmic factor. Then, we make use of techniques from online learning to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample complexity. Specifically, we cast GDRO as a two-players game where one player simply performs SMD and the other executes an online algorithm for non-oblivious multi-armed bandits. Next, we consider a more practical scenario where the number of samples that can be drawn from each distribution is different, and propose a novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates. Denote by $n_i$ the sample budget for the $i$-th distribution, and assume $n_1 \geq n_2 \geq \cdots \geq n_m$. In the first approach, we incorporate non-uniform sampling into SMD such that the sample budget is satisfied in expectation, and prove that the excess risk of the $i$-th distribution decreases at an $O(\sqrt{n_1 \log m}/n_i)$ rate. In the second approach, we use mini-batches to meet the budget exactly and also reduce the variance in stochastic gradients, and then leverage stochastic mirror-prox algorithm, which can exploit small variances, to optimize a carefully designed weighted GDRO problem. Under appropriate conditions, it attains an $O((\log m)/\sqrt{n_i})$ convergence rate, which almost matches the optimal $O(\sqrt{1/n_i})$ rate of only learning from the $i$-th distribution with $n_i$ samples. | cs | 0 |
Title: Correctness Comparison of ChatGPT-4, Bard, Claude-2, and Copilot for Spatial Tasks
Abstract: Generative AI including large language models (LLMs) have recently gained significant interest in the geo-science community through its versatile task-solving capabilities including coding, spatial computations, generation of sample data, time-series forecasting, toponym recognition, or image classification. So far, the assessment of LLMs for spatial tasks has primarily focused on ChatGPT, arguably the most prominent AI chatbot, whereas other chatbots received less attention. To narrow this research gap, this study evaluates the correctness of responses for a set of 54 spatial tasks assigned to four prominent chatbots, i.e., ChatGPT-4, Bard, Claude-2, and Copilot. Overall, the chatbots performed well on spatial literacy, GIS theory, and interpretation of programming code and given functions, but revealed weaknesses in mapping, code generation, and code translation. ChatGPT-4 outperformed other chatbots across most task categories. | cs | 0 |
Title: Solution of the Kolmogorov equation for TASEP
Abstract: We provide a direct and elementary proof that the formula obtained in [MQR17] for the TASEP transition probabilities for general (one-sided) initial data solves the Kolmogorov backward equation. The same method yields the solution for the related PushASEP particle system. | math | 1 |
Title: LLM Augmented LLMs: Expanding Capabilities through Composition
Abstract: Foundational models with billions of parameters which have been trained on large corpora of data have demonstrated non-trivial skills in a variety of domains. However, due to their monolithic structure, it is challenging and expensive to augment them or impart new skills. On the other hand, due to their adaptation abilities, several new instances of these models are being trained towards new domains and tasks. In this work, we study the problem of efficient and practical composition of existing foundation models with more specific models to enable newer capabilities. To this end, we propose CALM -- Composition to Augment Language Models -- which introduces cross-attention between models to compose their representations and enable new capabilities. Salient features of CALM are: (i) Scales up LLMs on new tasks by 're-using' existing LLMs along with a few additional parameters and data, (ii) Existing model weights are kept intact, and hence preserves existing capabilities, and (iii) Applies to diverse domains and settings. We illustrate that augmenting PaLM2-S with a smaller model trained on low-resource languages results in an absolute improvement of up to 13\% on tasks like translation into English and arithmetic reasoning for low-resource languages. Similarly, when PaLM2-S is augmented with a code-specific model, we see a relative improvement of 40\% over the base model for code generation and explanation tasks -- on-par with fully fine-tuned counterparts. | cs | 0 |
Title: LLaVA-$φ$: Efficient Multi-Modal Assistant with Small Language Model
Abstract: In this paper, we introduce LLaVA-$\phi$ (LLaVA-Phi), an efficient multi-modal assistant that harnesses the power of the recently advanced small language model, Phi-2, to facilitate multi-modal dialogues. LLaVA-Phi marks a notable advancement in the realm of compact multi-modal models. It demonstrates that even smaller language models, with as few as 2.7B parameters, can effectively engage in intricate dialogues that integrate both textual and visual elements, provided they are trained with high-quality corpora. Our model delivers commendable performance on publicly available benchmarks that encompass visual comprehension, reasoning, and knowledge-based perception. Beyond its remarkable performance in multi-modal dialogue tasks, our model opens new avenues for applications in time-sensitive environments and systems that require real-time interaction, such as embodied agents. It highlights the potential of smaller language models to achieve sophisticated levels of understanding and interaction, while maintaining greater resource efficiency.The project is available at {https://github.com/zhuyiche/llava-phi}. | cs | 0 |
Title: Actions of right-angled Artin groups in low dimensions
Abstract: We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional manifolds. For compact one--manifolds, every right-angled Artin group acts faithfully by $C^1$ diffeomorphisms, but the right-angled Artin groups which act faithfully by $C^2$ diffeomorphisms are very restricted. For the real line, every right-angled Artin group acts faithfully by $C^{\infty}$ diffeomorphisms, though analytic actions are again more limited. In dimensions two and higher, every right-angled Artin group acts faithfully on every manifold by $C^{\infty}$ diffeomorphisms. We give applications of this discussion to mapping class groups of surfaces and related groups. | math | 1 |
Title: Some combinatorial problems arising in the dimer model
Abstract: We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference. | math | 0 |
Title: Microlocal approach to Lusztig's symmetries
Abstract: We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves. | math | 1 |
Title: A fourth-order Cherrier-Escobar problem with prescribed corner behavior on the half-ball
Abstract: We show that the half-ball in $\mathbb{R}^4$ can be conformally changed so that the only contribution to the Gauss--Bonnet formula is a constant term at the corner. This may be seen as a fourth-order Cherrier--Escobar-type problem on the half-ball. | math | 0 |
Title: Act as You Learn: Adaptive Decision-Making in Non-Stationary Markov Decision Processes
Abstract: A fundamental (and largely open) challenge in sequential decision-making is dealing with non-stationary environments, where exogenous environmental conditions change over time. Such problems are traditionally modeled as non-stationary Markov decision processes (NSMDP). However, existing approaches for decision-making in NSMDPs have two major shortcomings: first, they assume that the updated environmental dynamics at the current time are known (although future dynamics can change); and second, planning is largely pessimistic, i.e., the agent acts ``safely'' to account for the non-stationary evolution of the environment. We argue that both these assumptions are invalid in practice -- updated environmental conditions are rarely known, and as the agent interacts with the environment, it can learn about the updated dynamics and avoid being pessimistic, at least in states whose dynamics it is confident about. We present a heuristic search algorithm called \textit{Adaptive Monte Carlo Tree Search (ADA-MCTS)} that addresses these challenges. We show that the agent can learn the updated dynamics of the environment over time and then act as it learns, i.e., if the agent is in a region of the state space about which it has updated knowledge, it can avoid being pessimistic. To quantify ``updated knowledge,'' we disintegrate the aleatoric and epistemic uncertainty in the agent's updated belief and show how the agent can use these estimates for decision-making. We compare the proposed approach with the multiple state-of-the-art approaches in decision-making across multiple well-established open-source problems and empirically show that our approach is faster and highly adaptive without sacrificing safety. | cs | 0 |
Title: Theory inspired deep network for instantaneous-frequency extraction and signal components recovery from discrete blind-source data
Abstract: This paper is concerned with the inverse problem of recovering the unknown signal components, along with extraction of their instantaneous frequencies (IFs), governed by the adaptive harmonic model (AHM), from discrete (and possibly non-uniform) samples of the blind-source composite signal. None of the existing decomposition methods and algorithms, including the most popular empirical mode decomposition (EMD) computational scheme and its current modifications, is capable of solving this inverse problem. In order to meet the AHM formulation and to extract the IFs of the decomposed components, called intrinsic mode functions (IMFs), each IMF of EMD is extended to an analytic function in the upper half of the complex plane via the Hilbert transform, followed by taking the real part of the polar form of the analytic extension. Unfortunately, this approach most often fails to resolve the inverse problem satisfactorily. More recently, to resolve the inverse problem, the notion of synchrosqueezed wavelet transform (SST) was proposed by Daubechies and Maes, and further developed in many other papers, while a more direct method, called signal separation operation (SSO), was proposed and developed in our previous work published in the journal, Applied and Computational Harmonic Analysis, vol. 30(2):243-261, 2016. In the present paper, we propose a synthesis of SSO using a deep neural network, based directly on a discrete sample set, that may be non-uniformly sampled, of the blind-source signal. Our method is localized, as illustrated by a number of numerical examples, including components with different signal arrival and departure times. It also yields short-term prediction of the signal components, along with their IFs. Our neural networks are inspired by theory, designed so that they do not require any training in the traditional sense. | cs | 1 |
Title: The Least Common Multiple of Polynomial Values over Function Fields
Abstract: Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $d \geq 2$ one has $$\log\left[\mathrm{lcm}(f(1),f(2),\ldots f(N))\right]\sim(d-1)N\log N$$ as $N \to \infty$. He proved it in the case $d=2$ but it remains open for every polynomial with $d>2$. We investigate the function field analogue of the problem by considering polynomials over the ring $\mathbb F_q[T]$. We state an analog of Cilleruelo's conjecture in this setting: denoting by $$L_f(n) := \mathrm{lcm} \left(f\left(Q\right)\ : \ Q \in \mathbb F_q[T]\mbox{ monic},\, \mathrm{deg}\,Q = n\right)$$ we conjecture that \begin{equation}\label{eq:conjffabs}\mathrm{deg}\, L_f(n) \sim c_f \left(d-1\right) nq^n,\ n \to \infty\end{equation} ($c_f$ is an explicit constant dependent only on $f$, typically $c_f=1$). We give both upper and lower bounds for $L_f(n)$ and show that the conjectured asymptotic holds for a class of ``special" polynomials, initially considered by Leumi in this context, which includes all quadratic polynomials and many other examples as well. We fully classify these special polynomials. We also show that $\mathrm{deg}\, L_f(n) \sim \mathrm{deg}\,\mathrm{rad}\left(L_f(n)\right)$ (in other words the corresponding LCM is close to being squarefree), which is not known over $\mathbb Z$. | math | 0 |
Title: A simple proof for generalized Fibonacci numbers with dying rabbits
Abstract: We consider the generalized Fibonacci counting problem with rabbits that become fertile at age $f$ and die at age $d$, with $1<=f<=d$ and $d$ finite or infinite. We provide a simple proof, based exclusively on a counting argumentation, for a recursive formula that gives the $n$th generalized Fibonacci number as a function of at most 3 previous numbers. The formula generalizes both the original Fibonacci sequence, for $f=2$ and $d=\infty$ (or $f=1$ and $d=2$), and other Fibonacci-related sequences, such as the Padovan sequence, for $f=2$ and $d=3$, the Tribonacci, for $f=1$ and $d=3$, Tetranacci, for $f=1$ and $d=4$, and alike sequences, for $f=1$ and finite values of $d$. | math | 0 |
Title: Travelers: A scalable fair ordering BFT system
Abstract: Many blockchain platform are subject to maximal value extraction (MEV), and users on the platform are losing money while sending transactions because the transaction order can be manipulated to extract value from them. Consensus protocols have been augmented with different notion of fair ordering in order to counter the problem. Out of all practical protocols, the most efficient BFT consensus requires $O(nTL + n^2T)$ communication complexity, where $n$ is number node, $T$ is number of transactions and $L$ is average transaction size. In this work, we propose a new system of BFT fair ordering protocols, Travelers, that substantially reduce the communication complexity. The proposed system of protocols satisfy a new notion of fair ordering, called probabilistic fair ordering, which is an extension to some existing notions of fairness. The new notion allows a small probability of error $\epsilon$, that adversary can insert some transactions at any location in a block, but for the remaining $1-\epsilon$ the a modified version of ordering linearizability holds. Our mechanism neither require a dissemination network nor direct submissions to all consensus nodes. The key innovation comes from a routing protocol, that is both flexible and efficient. We construct a protocol with $O(c\log({n})TL + n^2)$ communication complexity with $\epsilon = 1/n^c$ for some system parameter $c\ge 1$. | cs | 0 |
Title: Marked random graphs with given degree sequence: large deviations on the local topology
Abstract: We investigate the behavior of the empirical neighbourhood distribution of marked graphs in the framework of local weak convergence. We establish a large deviation principle for such families of empirical measures. The proof builds on Bordenave and Caputo's seminal 2015 paper, and Delgosha and Anantharam's 2019 introduction of BC entropy, relying on combinatorial lemmas that allow one to construct suitable approximations of measures supported on marked trees. | math | 0 |
Title: An alternative approach to large deviations for the almost-critical Erdős-Rényi random graph
Abstract: We study the near-critical behavior of the sparse Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$ on $n\gg1$ vertices, where the connection probability $p$ satisfies $np = 1+\theta(b_n^2/n)^{1/3}$, with $n^{3/10}\ll {b_n}\ll n^{1/2}$, and $\theta\in\mathbb{R}$. To this end, we introduce an empirical measure that describes connected components of $\mathcal{G}(n,p)$ of mesoscopic size $\propto (nb_n)^{2/3}$, and we characterize its large deviation behavior. The proof hinges on detailed combinatorial estimates and optimization procedures. In particular, we give precise estimates for the probability that the graph has no connected component of mesoscopic size or larger. We argue that these are a stepping stone for the analysis of more general inhomogeneous random graphs. Our proof strategy gives new and accurate estimates of the probability that the sparse Erd\H{o}s-R\'enyi graph is connected. | math | 0 |
Title: Multi-segmented non-isothermal compositional liquid gas well model for geothermal processes
Abstract: We consider a non-isothermal compositional gas liquid model for the simulation of well operations in geothermal processes. The model accounts for phase transitions assumed to be at thermodynamical equilibrium and is based on an hydrodynamical Drift Flux Model (DFM) combined with a No Pressure Wave approximation of the momentum equation. The focus of this work is on the design of a robust discretization accounting for slanted and multibranch wells with the ability to simulate both transient behavior such as well opening as well as coupled simulations at the time scale of the reservoir. It is based on a staggered finite volume scheme in space combined with a fully implicit Euler time integration. The construction of consistent and stable numerical fluxes is a key feature for a robust numerical method. It is achieved by combining a monotone flux approximation for the phase superficial velocities with an upwind approximation of the phase molar fractions, density and enthalpy. In order to facilitate the coupling of the well and reservoir models, the Newton linearization accounts for the elimination of the hydrodynamical unknowns leading to Jacobian systems using the same primary unknowns than those of the reservoir model. The efficiency of our approach is investigated on both stand alone well test cases without and with cross flow, and on a fully coupled well-reservoir simulation. | math | 0 |
Title: A study in sums of products
Abstract: We give a general version of cancellation in exponential sums that arise as sums of products of trace functions satisfying a suitable independence condition related to the Goursat-Kolchin-Ribet criterion, in a form that is easily applicable in analytic number theory. | math | 1 |