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Title: Cambrian triangulations and their tropical realizations
Abstract: This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical realizations. For any signature $\varepsilon \in \{\pm\}^n$, we consider a family of $\varepsilon$-trees in bijection with the triangulations of the $\varepsilon$-polygon. These $\varepsilon$-trees define a flag regular triangulation $\mathcal{T}^\varepsilon$ of the subpolytope $\operatorname{conv} \{(\mathbf{e}_{i_\bullet}, \mathbf{e}_{j_\circ}) \, | \, 0 \le i_\bullet < j_\circ \le n+1 \}$ of the product of simplices $\triangle_{\{0_\bullet, \dots, n_\bullet\}} \times \triangle_{\{1_\circ, \dots, (n+1)_\circ\}}$. The oriented dual graph of the triangulation $\mathcal{T}^\varepsilon$ is the Hasse diagram of the (type $A$) $\varepsilon$-Cambrian lattice of N. Reading. For any $I_\bullet \subseteq \{0_\bullet, \dots, n_\bullet\}$ and $J_\circ \subseteq \{1_\circ, \dots, (n+1)_\circ\}$, we consider the restriction $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ of the triangulation $\mathcal{T}^\varepsilon$ to the face $\triangle_{I_\bullet} \times \triangle_{J_\circ}$. Its dual graph is naturally interpreted as the increasing flip graph on certain $(\varepsilon, I_\bullet, J_\circ)$-trees, which is shown to be a lattice generalizing in particular the $\nu$-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ as a polyhedral complex induced by a tropical hyperplane arrangement. | math | 1 |
Title: Three term rational function progressions in finite fields
Abstract: Let $F(t),G(t)\in \mathbb{Q}(t)$ be rational functions such that $F(t),G(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the three term rational function progressions of the form $x,x+F(y),x+G(y)$ in subsets of $\mathbb{F}_p$. The main new ingredient is an algebraic geometry version of PET induction that bypasses Weyl's differencing. This answers a question of Bourgain and Chang. | math | 0 |
Title: Quantiles on global non-positive curvature spaces
Abstract: This paper develops a notion of geometric quantiles on Hadamard spaces, also known as global non-positive curvature spaces. After providing some definitions and basic properties, including scaled isometry equivariance and a necessary condition on the gradient of the quantile loss function at quantiles on Hadamard manifolds, we investigate asymptotic properties of sample quantiles on Hadamard manifolds, such as strong consistency and joint asymptotic normality. We provide a detailed description of how to compute quantiles using a gradient descent algorithm in hyperbolic space and, in particular, an explicit formula for the gradient of the quantile loss function, along with experiments using simulated and real single-cell RNA sequencing data. | math | 0 |
Title: Spatiotemporal Attention for Multivariate Time Series Prediction and Interpretation
Abstract: Multivariate time series modeling and prediction problems are abundant in many machine learning application domains. Accurate interpretation of such prediction outcomes from a machine learning model that explicitly captures temporal correlations can significantly benefit the domain experts. In this context, temporal attention has been successfully applied to isolate the important time steps for the input time series. However, in multivariate time series problems, spatial interpretation is also critical to understand the contributions of different variables on the model outputs. We propose a novel deep learning architecture, called spatiotemporal attention mechanism (STAM) for simultaneous learning of the most important time steps and variables. STAM is a causal (i.e., only depends on past inputs and does not use future inputs) and scalable (i.e., scales well with an increase in the number of variables) approach that is comparable to the state-of-the-art models in terms of computational tractability. We demonstrate our models' performance on two popular public datasets and a domain-specific dataset. When compared with the baseline models, the results show that STAM maintains state-of-the-art prediction accuracy while offering the benefit of accurate spatiotemporal interpretability. The learned attention weights are validated from a domain knowledge perspective for these real-world datasets. | cs | 1 |
Title: Symplectic period for a representation of $GL_n(D)$
Abstract: In this paper, we study the irreducible admissible representations of $GL_{n}(D)$ with a symplectic period for $n = 3$ and $4$, i.e., those representations which have a linear functional invariant under $Sp_n(D)$, where $D$ is a quaternion division algebra over a non-archimedean local field $k$ of characteristic zero and $Sp_n(D)$ is the unique non-split inner form of the symplectic group $Sp_{2n}(k)$. Our results contain all distinguished unitary representations stated in Prasad's conjecture. | math | 0 |
Title: First International HARTING Open Source Prize Winner: The igus Humanoid Open Platform
Abstract: The use of standard platforms in the field of humanoid robotics can lower the entry barrier for new research groups, and accelerate research by the facilitation of code sharing. Numerous humanoid standard platforms exist in the lower size ranges of up to 60cm, but beyond that humanoid robots scale up quickly in weight and price, becoming less affordable and more difficult to operate, maintain and modify. The igus Humanoid Open Platform is an affordable, fully open-source platform for humanoid research. At 92cm, the robot is capable of acting in an environment meant for humans, and is equipped with enough sensors, actuators and computing power to support researchers in many fields. The structure of the robot is entirely 3D printed, leading to a lightweight and visually appealing design. This paper covers the mechanical and electrical aspects of the robot, as well as the main features of the corresponding open-source ROS software. At RoboCup 2016, the platform was awarded the first International HARTING Open Source Prize. | cs | 1 |
Title: Reconstructing almost all of a point set in $\mathbb{R}^d$ from randomly revealed pairwise distances
Abstract: Let $V$ be a set of $n$ points in $\mathbb{R}^d$, and suppose that the distance between each pair of points is revealed independently with probability $p$. We study when this information is sufficient to reconstruct large subsets of $V$, up to isometry. Strong results for $d=1$ have been obtained by Gir\~ao, Illingworth, Michel, Powierski, and Scott. In this paper, we investigate higher dimensions, and show that if $p>n^{-2/(d+4)}$, then we can reconstruct almost all of $V$ up to isometry, with high probability. We do this by relating it to a polluted graph bootstrap percolation result, for which we adapt the methods of Balogh, Bollob\'as, and Morris. | math | 0 |
Title: A New Criterion on Normal Bases of Finite Field Extensions
Abstract: A new criterion on normal bases of finite field extension $\mathbb{F}_{q^n} / \mathbb{F}_{q}$ is presented and explicit criterions for several particular finite field extensions are derived from this new criterion. | math | 1 |
Title: Non-commutative ambits and equivariant compactifications
Abstract: We prove that an action $\rho:A\to M(C_0(\mathbb{G})\otimes A)$ of a locally compact quantum group on a $C^*$-algebra has a universal equivariant compactification, and prove a number of other category-theoretic results on $\mathbb{G}$-equivariant compactifications: that the categories compactifications of $\rho$ and $A$ respectively are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When $\mathbb{G}$ is regular coamenable we also show that the forgetful functor from unital $\mathbb{G}$-$C^*$-algebras to unital $C^*$-algebras creates finite limits and is comonadic, and that the monomorphisms in the former category are injective. | math | 1 |
Title: Minimum $2$-vertex strongly biconnected spanning directed subgraph problem
Abstract: A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ is strongly biconnected for every vertex $w\in V$. In this paper we study the following problem. Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected. | cs | 1 |
Title: Toward 6G: From New Hardware Design to Wireless Semantic and Goal-Oriented Communication Paradigms
Abstract: Several speculative visions are conjecturing on what 6G services will be able to offer at the horizon of 2030. Nevertheless, the 6G design process is at its preliminary stages. The reality today is that hardware, technologies and new materials required to effectively meet the unprecedented performance targets required for future 6G services and network operation, have not been designed, tested or even do not exist yet. Today, a solid vision on the cost-benefit trade-offs of machine learning and artificial intelligence support for 6G network and services operation optimization is missing. This includes the possible support from hardware efficiency, operation effectiveness and, the immeasurable cost due to data acquisition-transfer-processing. The contribution of this paper is three-fold. This is the first paper deriving crucial 6G key performance indicators on hardware and technology design. Second, we present a new hardware technologies design methodology conceived to enable the effective software-hardware components integration required to meet the challenging performance envisioned for future 6G networks. Third, we suggest a paradigm shift towards goal-oriented and semantic communications, in which a totally new opportunity of joint design of hardware, artificial intelligence and effective communication is offered. The proposed vision is consolidated by our recent results on hardware, technology and machine learning performance. | cs | 1 |
Title: Differentially Private Sketches for Jaccard Similarity Estimation
Abstract: This paper describes two locally-differential private algorithms for releasing user vectors such that the Jaccard similarity between these vectors can be efficiently estimated. The basic building block is the well known MinHash method. To achieve a privacy-utility trade-off, MinHash is extended in two ways using variants of Generalized Randomized Response and the Laplace Mechanism. A theoretical analysis provides bounds on the absolute error and experiments show the utility-privacy trade-off on synthetic and real-world data. The paper ends with a critical discussion of related work. | cs | 1 |
Title: Introducing Packet-Level Analysis in Programmable Data Planes to Advance Network Intrusion Detection
Abstract: Programmable data planes offer precise control over the low-level processing steps applied to network packets, serving as a valuable tool for analysing malicious flows in the field of intrusion detection. Albeit with limitations on physical resources and capabilities, they allow for the efficient extraction of detailed traffic information, which can then be utilised by Machine Learning (ML) algorithms responsible for identifying security threats. In addressing resource constraints, existing solutions in the literature rely on compressing network data through the collection of statistical traffic features in the data plane. While this compression saves memory resources in switches and minimises the burden on the control channel between the data and the control plane, it also results in a loss of information available to the Network Intrusion Detection System (NIDS), limiting access to packet payload, categorical features, and the semantic understanding of network communications, such as the behaviour of packets within traffic flows. This paper proposes P4DDLe, a framework that exploits the flexibility of P4-based programmable data planes for packet-level feature extraction and pre-processing. P4DDLe leverages the programmable data plane to extract raw packet features from the network traffic, categorical features included, and to organise them in a way that the semantics of traffic flows are preserved. To minimise memory and control channel overheads, P4DDLe selectively processes and filters packet-level data, so that only the features required by the NIDS are collected. The experimental evaluation with recent Distributed Denial of Service (DDoS) attack data demonstrates that the proposed approach is very efficient in collecting compact and high-quality representations of network flows, ensuring precise detection of DDoS attacks. | cs | 0 |
Title: Faster optimal univariate microgaggregation
Abstract: Microaggregation is a method to coarsen a dataset, by optimally clustering data points in groups of at least $k$ points, thereby providing a $k$-anonymity type disclosure guarantee for each point in the dataset. Previous algorithms for univariate microaggregation had a $O(k n)$ time complexity. By rephrasing microaggregation as an instance of the concave least weight subsequence problem, in this work we provide improved algorithms that provide an optimal univariate microaggregation on sorted data in $O(n)$ time and space. We further show that our algorithms work not only for sum of squares cost functions, as typically considered, but seamlessly extend to many other cost functions used for univariate microaggregation tasks. In experiments we show that the presented algorithms lead to real world performance improvements. | cs | 0 |
Title: DHOT-GM: Robust Graph Matching Using A Differentiable Hierarchical Optimal Transport Framework
Abstract: Graph matching is one of the most significant graph analytic tasks in practice, which aims to find the node correspondence across different graphs. Most existing approaches rely on adjacency matrices or node embeddings when matching graphs, whose performances are often sub-optimal because of not fully leveraging the multi-modal information hidden in graphs, such as node attributes, subgraph structures, etc. In this study, we propose a novel and effective graph matching method based on a differentiable hierarchical optimal transport (HOT) framework, called DHOT-GM. Essentially, our method represents each graph as a set of relational matrices corresponding to the information of different modalities. Given two graphs, we enumerate all relational matrix pairs and obtain their matching results, and accordingly, infer the node correspondence by the weighted averaging of the matching results. This method can be implemented as computing the HOT distance between the two graphs -- each matching result is an optimal transport plan associated with the Gromov-Wasserstein (GW) distance between two relational matrices, and the weights of all matching results are the elements of an upper-level optimal transport plan defined on the matrix sets. We propose a bi-level optimization algorithm to compute the HOT distance in a differentiable way, making the significance of the relational matrices adjustable. Experiments on various graph matching tasks demonstrate the superiority and robustness of our method compared to state-of-the-art approaches. | cs | 0 |
Title: Higher Descent Data as a Homotopy Limit
Abstract: We define the 2-groupoid of descent data assigned to a cosimplicial 2-groupoid and present it as the homotopy limit of the cosimplicial space gotten after applying the 2-nerve in each cosimplicial degree. This can be applied also to the case of $n$-groupoids thus providing an analogous presentation of "descent data" in higher dimensions. | math | 1 |
Title: On the Theoretical Gap of Channel Hopping Sequences with Maximum Rendezvous Diversity in the Multichannel Rendezvous Problem
Abstract: In the literature, there are several well-known periodic channel hopping (CH) sequences that can achieve maximum rendezvous diversity in a cognitive radio network (CRN). For a CRN with $N$ channels, it is known that the period of such a CH sequence is at least $N^2$. The asymptotic approximation ratio, defined as the ratio of the period of a CH sequence to the lower bound $N^2$ when $N \to \infty$, is still 2.5 for the best known CH sequence in the literature. An open question in the multichannel rendezvous problem is whether it is possible to construct a periodic CH sequence that has the asymptotic approximation ratio 1. In this paper, we tighten the theoretical gap by proposing CH sequences, called IDEAL-CH, that have the asymptotic approximation ratio 2. For a weaker requirement that only needs the two users to rendezvous on one commonly available channel in a period, we propose channel hopping sequences, called ORTHO-CH, with period $(2p +1)p$, where $p$ is the smallest prime not less than $N$. | cs | 1 |
Title: Some Fibonacci-Related Sequences
Abstract: We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the Walnut theorem-prover. We also prove two new results about synchronized sequences. | math | 0 |
Title: On instability of Type (II) Lawson-Osserman Cones
Abstract: We obtain the instability of Type (II) Lawson-Osserman cones in Euclidean spaces, and thus provide a family of (uncountably many) unstable solutions with singularity to the Dirichlet problem for minimal graphs of high codimension versus smooth unstable ones by Lawson-Osserman through a min-max technique. To our knowledge, these are the first examples of non-smooth unstable minimal graphs and unlikely detectible through the mean curvature flow or min-max theory. | math | 1 |
Title: Ramsey Partial Orders from Acyclic Graphs
Abstract: We prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that class of acyclic graphs has the Ramsey property and uses the partite construction. | math | 1 |
Title: On a reduction map for Drinfeld modules
Abstract: In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehat\varphi}={\phi}_{1}^{e_1}\times \dots \times {\phi}_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assume that the ${\mathrm{rank}}(\phi)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\mathrm{End}}({\phi}_{i})=A$ for $ i=1,\dots , t.$ Our main result is the following numeric criterion. Let ${N}={N}_{1}^{e_1}\times\dots\times {N}_{t}^{e_t}$ be a finitely generated $A$ submodule of the Mordell-Weil group ${\widehat\varphi}({\cal O}_{K})={\phi}_{1}({\cal O}_{K})^{e_{1}}\times\dots\times {\phi}_{t}({\cal O}_{K})^{{e}_{t}},$ and let ${\Lambda}\subset N$ be an $A$ - submodule. If we assume $d_{i}\geq e_{i}$ and $P\in N$ such that $r_{\cal W}(P)\in r_{\cal W}({\Lambda}) $ for almost all primes ${\cal W}$ of ${\cal O}_{K},$ then $P\in {\Lambda}+N_{tor}.$ We also build on the recent results of S.Bara{\'n}czuk \cite{b17} concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned $t$-modules. | math | 1 |
Title: Murre's conjectures for certain product varieties
Abstract: We consider Murre's conjectures on Chow groups for a fourfold which is a product of two curves and a surface. We give a result which concerns Conjecture D:the kernel of a certain projector is equal to the homologically trivial part of the Chow group. We also give a proof of Conjecture B for a product of two surfaces. | math | 1 |
Title: Approximate Approximations from scattered data
Abstract: The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators. | math | 1 |
Title: Counterexamples to the local-global principle associated with Swinnerton-Dyer's cubic form
Abstract: In this paper, we imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematica (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of $4, 5, 6, ..., 2n+2$ variables of degree $2n+1$ for infinitely many integers $n$. It is contrastive to Swinnerton-Dyer's original construction that we do not need any concrete calculation in the proof of local solubility. | math | 1 |
Title: On the linear complexity for multidimensional sequences
Abstract: In this paper, we define the linear complexity for multidimensional sequences over finite fields, generalizing the one-dimensional case. We give some lower and upper bounds, valid with large probability, for the linear complexity and $k$-error linear complexity of multidimensional periodic sequences. | math | 1 |
Title: PEGASUS: Physically Enhanced Gaussian Splatting Simulation System for 6DOF Object Pose Dataset Generation
Abstract: We introduce Physically Enhanced Gaussian Splatting Simulation System (PEGASUS) for 6DOF object pose dataset generation, a versatile dataset generator based on 3D Gaussian Splatting. Environment and object representations can be easily obtained using commodity cameras to reconstruct with Gaussian Splatting. PEGASUS allows the composition of new scenes by merging the respective underlying Gaussian Splatting point cloud of an environment with one or multiple objects. Leveraging a physics engine enables the simulation of natural object placement within a scene through interaction between meshes extracted for the objects and the environment. Consequently, an extensive amount of new scenes - static or dynamic - can be created by combining different environments and objects. By rendering scenes from various perspectives, diverse data points such as RGB images, depth maps, semantic masks, and 6DoF object poses can be extracted. Our study demonstrates that training on data generated by PEGASUS enables pose estimation networks to successfully transfer from synthetic data to real-world data. Moreover, we introduce the Ramen dataset, comprising 30 Japanese cup noodle items. This dataset includes spherical scans that captures images from both object hemisphere and the Gaussian Splatting reconstruction, making them compatible with PEGASUS. | cs | 0 |
Title: Real-Time 2D Temperature Field Prediction in Metal Additive Manufacturing Using Physics-Informed Neural Networks
Abstract: Accurately predicting the temperature field in metal additive manufacturing (AM) processes is critical to preventing overheating, adjusting process parameters, and ensuring process stability. While physics-based computational models offer precision, they are often time-consuming and unsuitable for real-time predictions and online control in iterative design scenarios. Conversely, machine learning models rely heavily on high-quality datasets, which can be costly and challenging to obtain within the metal AM domain. Our work addresses this by introducing a physics-informed neural network framework specifically designed for temperature field prediction in metal AM. This framework incorporates a physics-informed input, physics-informed loss function, and a Convolutional Long Short-Term Memory (ConvLSTM) architecture. Utilizing real-time temperature data from the process, our model predicts 2D temperature fields for future timestamps across diverse geometries, deposition patterns, and process parameters. We validate the proposed framework in two scenarios: full-field temperature prediction for a thin wall and 2D temperature field prediction for cylinder and cubic parts, demonstrating errors below 3% and 1%, respectively. Our proposed framework exhibits the flexibility to be applied across diverse scenarios with varying process parameters, geometries, and deposition patterns. | cs | 0 |
Title: A macroscopic model for a system of swarming agents using curvature control
Abstract: In this paper, we study the macroscopic limit of a new model of collective displacement. The model, called PTWA, is a combination of the Vicsek alignment model and the Persistent Turning Walker (PTW) model of motion by curvature control. The PTW model was designed to fit measured trajectories of individual fish. The PTWA model (Persistent Turning Walker with Alignment) describes the displacements of agents which modify their curvature in order to align with their neighbors. The derivation of its macroscopic limit uses the non-classical notion of generalized collisional invariant. The macroscopic limit of the PTWA model involves two physical quantities, the density and the mean velocity of individuals. It is a system of hyperbolic type but is non-conservative due to a geometric constraint on the velocity. This system has the same form as the macroscopic limit of the Vicsek model (the 'Vicsek hydrodynamics') but for the expression of the model coefficients. The numerical computations show that the numerical values of the coefficients are very close. The 'Vicsek Hydrodynamic model' appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated. | math | 1 |
Title: $F$-finiteness of homomorphisms and its descent
Abstract: Let $p$ be a prime number. We define the notion of $F$-finiteness of homomorphisms of $\mathbb F_p$-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on $F$-finiteness of homomorphisms of $\mathbb F_p$-algebras. As a corollary, we prove the following. Let $g:B\to C$ be a homomorphism of Noetherian $\mathbb F_p$-algebras. If $g$ is faithfully flat reduced, and $C$ is $F$-finite, then $B$ is $F$-finite. This is a generalization of Seydi's result on excellent local rings of characteristic $p$. | math | 1 |
Title: Improving Automated Program Repair with Domain Adaptation
Abstract: Automated Program Repair (APR) is defined as the process of fixing a bug/defect in the source code, by an automated tool. APR tools have recently experienced promising results by leveraging state-of-the-art Neural Language Processing (NLP) techniques. APR tools such as TFix and CodeXGLUE combine text-to-text transformers with software-specific techniques are outperforming alternatives, these days. However, in most APR studies the train and test sets are chosen from the same set of projects. In reality, however, APR models are meant to be generalizable to new and different projects. Therefore, there is a potential threat that reported APR models with high effectiveness perform poorly when the characteristics of the new project or its bugs are different than the training set's(Domain Shift). In this study, we first define and measure the domain shift problem in automated program repair. Then, we then propose a domain adaptation framework that can adapt an APR model for a given target project. We conduct an empirical study with three domain adaptation methods FullFineTuning, TuningWithLightWeightAdapterLayers, and CurriculumLearning using two state-of-the-art domain adaptation tools (TFix and CodeXGLUE) and two APR models on 611 bugs from 19 projects. The results show that our proposed framework can improve the effectiveness of TFix by 13.05% and CodeXGLUE by 23.4%. Another contribution of this study is the proposal of a data synthesis method to address the lack of labelled data in APR. We leverage transformers to create a bug generator model. We use the generated synthetic data to domain adapt TFix and CodeXGLUE on the projects with no data (Zero-shot learning), which results in an average improvement of 5.76% and 24.42% for TFix and CodeXGLUE, respectively. | cs | 1 |
Title: Learning Knowledge Representation with Meta Knowledge Distillation for Single Image Super-Resolution
Abstract: Knowledge distillation (KD), which can efficiently transfer knowledge from a cumbersome network (teacher) to a compact network (student), has demonstrated its advantages in some computer vision applications. The representation of knowledge is vital for knowledge transferring and student learning, which is generally defined in hand-crafted manners or uses the intermediate features directly. In this paper, we propose a model-agnostic meta knowledge distillation method under the teacher-student architecture for the single image super-resolution task. It provides a more flexible and accurate way to help the teachers transmit knowledge in accordance with the abilities of students via knowledge representation networks (KRNets) with learnable parameters. In order to improve the perception ability of knowledge representation to students' requirements, we propose to solve the transformation process from intermediate outputs to transferred knowledge by employing the student features and the correlation between teacher and student in the KRNets. Specifically, the texture-aware dynamic kernels are generated and then extract texture features to be improved and the corresponding teacher guidance so as to decompose the distillation problem into texture-wise supervision for further promoting the recovery quality of high-frequency details. In addition, the KRNets are optimized in a meta-learning manner to ensure the knowledge transferring and the student learning are beneficial to improving the reconstructed quality of the student. Experiments conducted on various single image super-resolution datasets demonstrate that our proposed method outperforms existing defined knowledge representation related distillation methods, and can help super-resolution algorithms achieve better reconstruction quality without introducing any inference complexity. | cs | 1 |
Title: Index theory for traveling waves in reaction diffusion systems with skew gradient structure
Abstract: A unified geometric approach for the stability analysis of traveling pulse solutions for reaction-diffusion equations with skew-gradient structure has been established in a previous paper [9], but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equation. | math | 1 |
Title: Generating synthetic data for neural operators
Abstract: Numerous developments in the recent literature show the promising potential of deep learning in obtaining numerical solutions to partial differential equations (PDEs) beyond the reach of current numerical solvers. However, data-driven neural operators all suffer from the same problem: the data needed to train a network depends on classical numerical solvers such as finite difference or finite element, among others. In this paper, we propose a new approach to generating synthetic functional training data that does not require solving a PDE numerically. The way we do this is simple: we draw a large number $N$ of independent and identically distributed `random functions' $u_j$ from the underlying solution space (e.g., $H_0^1(\Omega)$) in which we know the solution lies according to classical theory. We then plug each such random candidate solution into the equation and get a corresponding right-hand side function $f_j$ for the equation, and consider $(f_j, u_j)_{j=1}^N$ as supervised training data for learning the underlying inverse problem $f \rightarrow u$. This `backwards' approach to generating training data only requires derivative computations, in contrast to standard `forward' approaches, which require a numerical PDE solver, enabling us to generate a large number of such data points quickly and efficiently. While the idea is simple, we hope that this method will expand the potential for developing neural PDE solvers that do not depend on classical numerical solvers. | cs | 0 |
Title: Online minimum search for Brownian motion and the Cauchy process: Multiple approache
Abstract: The distribution for the minimum of Brownian motion or the Cauchy process is well-known using the reflection principle. Here we consider the problem of finding the sample-by-sample minimum, which we call the online minimum search. We consider the possibility of the golden search method, but we show quantitatively that the bisection method is more efficient. In the bisection method there is a hierarchical parameter, which tunes the depth to which each sub-search is conducted, somewhat similarly to how a depth-first search works to generate a topological ordering on nodes. Finally, we consider the possibility of using harmonic measure, which is a novel idea that has so far been unexplored. | math | 0 |
Title: Presentations of Kauffman bracket skein algebras of planar surfaces
Abstract: Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$, and suppose $q+q^{-1}$ is invertible in $R$. For each planar surface $\Sigma_{0,n+1}$, we present its Kauffman bracket skein algebra over $R$ by explicit generators and relations. The presentation is independent of $R$, and can be considered as a quantization of the trace algebra of $n$ generic $2\times 2$ unimodular matrices. | math | 0 |
Title: Globalizing and stabilizing global $\infty$-categories
Abstract: We consider the question of cocompleting partially presentable parametrized $\infty$-categories in the sense of arXiv:2307.11001. As our main result we show that in certain cases one may compute such relative cocompletions via a very explicit formula given in terms of partially lax limits. We then apply this to equivariant homotopy theory, building on the work of op. cit. and arXiv:2301.08240, to conclude that the global $\infty$-category of globally equivariant spectra is the relative cocompletion of the global $\infty$-category of equivariant spectra. Evaluating at a group $G$ we obtain a description of the $\infty$-category of $G$-global spectra as a partially lax limit, extending the main result of arXiv:2206.01556 for finite groups to $G$-global homotopy theory. Finally we investigate the question of stabilizing global $\infty$-categories by inverting the action of representation spheres, and deduce a second universal property for the global $\infty$-category of globally equivariant spectra, similar to that of arXiv:2302.06207. | math | 0 |
Title: Know Your Limits: Uncertainty Estimation with ReLU Classifiers Fails at Reliable OOD Detection
Abstract: A crucial requirement for reliable deployment of deep learning models for safety-critical applications is the ability to identify out-of-distribution (OOD) data points, samples which differ from the training data and on which a model might underperform. Previous work has attempted to tackle this problem using uncertainty estimation techniques. However, there is empirical evidence that a large family of these techniques do not detect OOD reliably in classification tasks. This paper gives a theoretical explanation for said experimental findings and illustrates it on synthetic data. We prove that such techniques are not able to reliably identify OOD samples in a classification setting, since their level of confidence is generalized to unseen areas of the feature space. This result stems from the interplay between the representation of ReLU networks as piece-wise affine transformations, the saturating nature of activation functions like softmax, and the most widely-used uncertainty metrics. | cs | 1 |
Title: Dynamic Local Regret for Non-convex Online Forecasting
Abstract: We consider online forecasting problems for non-convex machine learning models. Forecasting introduces several challenges such as (i) frequent updates are necessary to deal with concept drift issues since the dynamics of the environment change over time, and (ii) the state of the art models are non-convex models. We address these challenges with a novel regret framework. Standard regret measures commonly do not consider both dynamic environment and non-convex models. We introduce a local regret for non-convex models in a dynamic environment. We present an update rule incurring a cost, according to our proposed local regret, which is sublinear in time T. Our update uses time-smoothed gradients. Using a real-world dataset we show that our time-smoothed approach yields several benefits when compared with state-of-the-art competitors: results are more stable against new data; training is more robust to hyperparameter selection; and our approach is more computationally efficient than the alternatives. | cs | 1 |
Title: Galois subspaces for smooth projective curves
Abstract: Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to\mathbb{P}^1$. For genus $g\geq2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bundles over \'etale quotients of the elliptic curve, and we describe these components explicitly. | math | 1 |
Title: Pseudorandom Hashing for Space-bounded Computation with Applications in Streaming
Abstract: We revisit Nisan's classical pseudorandom generator (PRG) for space-bounded computation (STOC 1990) and its applications in streaming algorithms. We describe a new generator, HashPRG, that can be thought of as a symmetric version of Nisan's generator over larger alphabets. Our generator allows a trade-off between seed length and the time needed to compute a given block of the generator's output. HashPRG can be used to obtain derandomizations with much better update time and \emph{without sacrificing space} for a large number of data stream algorithms, such as $F_p$ estimation in the parameter regimes $p > 2$ and $0 < p < 2$ and CountSketch with tight estimation guarantees as analyzed by Minton and Price (SODA 2014) which assumed access to a random oracle. We also show a recent analysis of Private CountSketch can be derandomized using our techniques. For a $d$-dimensional vector $x$ being updated in a turnstile stream, we show that $\|x\|_{\infty}$ can be estimated up to an additive error of $\varepsilon\|x\|_{2}$ using $O(\varepsilon^{-2}\log(1/\varepsilon)\log d)$ bits of space. Additionally, the update time of this algorithm is $O(\log 1/\varepsilon)$ in the Word RAM model. We show that the space complexity of this algorithm is optimal up to constant factors. However, for vectors $x$ with $\|x\|_{\infty} = \Theta(\|x\|_{2})$, we show that the lower bound can be broken by giving an algorithm that uses $O(\varepsilon^{-2}\log d)$ bits of space which approximates $\|x\|_{\infty}$ up to an additive error of $\varepsilon\|x\|_{2}$. We use our aforementioned derandomization of the CountSketch data structure to obtain this algorithm, and using the time-space trade off of HashPRG, we show that the update time of this algorithm is also $O(\log 1/\varepsilon)$ in the Word RAM model. | cs | 0 |
Title: Random walk on the high-dimensional IIC
Abstract: We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation. | math | 1 |
Title: Matroid Products in Tropical Geometry
Abstract: Symmetric powers of matroids were first introduced by Lovasz and Mason in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor closed and has infinitely many forbidden minors. | math | 0 |
Title: Permawound Unipotent Groups
Abstract: We introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain "ubiquity" and "rigidity" properties that in combination render them very useful in the study of general wound unipotent groups. As an illustration of their utility, we present two applications: We prove that nonsplit smooth unipotent groups over (infinite) finitely-generated fields have infinite first cohomology; and we show that every commutative p-torsion wound unipotent group over a field of degree of imperfection 1 is the maximal unipotent quotient of a commutative pseudo-reductive group, thus partially answering a question of Totaro. | math | 0 |
Title: Linking forms of amphichiral knots
Abstract: We give a simple obstruction for a knot to be amphichiral, in terms of the homology of the 2-fold branched cover. We work with unoriented knots, and so obstruct both positive and negative amphichirality. | math | 1 |
Title: Beck modules and alternative algebras
Abstract: We set out the general theory of "Beck modules" in a variety of algebras and describe them as modules over suitable "universal enveloping" unital associative algebras. We pay particular attention to the somewhat nonstandard case of "alternative algebras," defined by a restricted associative law, and determine the Poincar\'e polynomial of the universal enveloping algebra in the homogenous case. | math | 0 |
Title: Detecting Hostile Posts using Relational Graph Convolutional Network
Abstract: This work is based on the submission to the competition Hindi Constraint conducted by AAAI@2021 for detection of hostile posts in Hindi on social media platforms. Here, a model is presented for detection and classification of hostile posts and further classify into fake, offensive, hate and defamation using Relational Graph Convolutional Networks. Unlike other existing work, our approach is focused on using semantic meaning along with contextutal information for better classification. The results from AAAI@2021 indicates that the proposed model is performing at par with Google's XLM-RoBERTa on the given dataset. Our best submission with RGCN achieves an F1 score of 0.97 (7th Rank) on coarse-grained evaluation and achieved best performance on identifying fake posts. Among all submissions to the challenge, our classification system with XLM-Roberta secured 2nd rank on fine-grained classification. | cs | 1 |
Title: Smoothing cones over K3 surfaces
Abstract: We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g\le 10 or g=12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone is smoothable even though the projective cone is not. | math | 1 |
Title: Non-isogenous superelliptic jacobians II
Abstract: Let $\ell$ be an odd prime and $K$ a field of characteristic different from $\ell$. Let $\bar{K}$ be an algebraic closure of $K$. Assume that $K$ contains a primitive $\ell$th root of unity. Let $n \ne \ell$ be another odd prime. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider superelliptic curves $C_{f,\ell}: y^{\ell}=f(x)$ and $C_{h,\ell}: y^{\ell}=h(x)$ of genus $(n-1)(\ell-1)/2$, and their jacobians $J^{(f,\ell)}$ and $J^{(h,\ell)}$, which are $(n-1)(\ell-1)/2$-dimensional abelian varieties over $\bar{K}$. Suppose that one of the polynomials is irreducible and the other reducible over $K$. We prove that if $J^{(f,\ell)}$ and $J^{(h,\ell)}$ are isogenous over $\bar{K}$ then both endomorphism algebras $\mathrm{End}^{0}(J^{(f,\ell)})$ and $\mathrm{End}^{0}(J^{(h,\ell)})$ contain an invertible element of multiplicative order $n$. | math | 0 |
Title: Structural properties of weak cotype 2 spaces
Abstract: Several characterizations of weak cotype 2 and weak Hilbert spaces are given in terms of basis constants and other structural invariants of Banach spaces. For finite-dimensional spaces, characterizations depending on subspaces of fixed proportional dimension are proved. | math | 1 |
Title: Pre-trained Recommender Systems: A Causal Debiasing Perspective
Abstract: Recent studies on pre-trained vision/language models have demonstrated the practical benefit of a new, promising solution-building paradigm in AI where models can be pre-trained on broad data describing a generic task space and then adapted successfully to solve a wide range of downstream tasks, even when training data is severely limited (e.g., in zero- or few-shot learning scenarios). Inspired by such progress, we investigate in this paper the possibilities and challenges of adapting such a paradigm to the context of recommender systems, which is less investigated from the perspective of pre-trained model. In particular, we propose to develop a generic recommender that captures universal interaction patterns by training on generic user-item interaction data extracted from different domains, which can then be fast adapted to improve few-shot learning performance in unseen new domains (with limited data). However, unlike vision/language data which share strong conformity in the semantic space, universal patterns underlying recommendation data collected across different domains (e.g., different countries or different E-commerce platforms) are often occluded by both in-domain and cross-domain biases implicitly imposed by the cultural differences in their user and item bases, as well as their uses of different e-commerce platforms. As shown in our experiments, such heterogeneous biases in the data tend to hinder the effectiveness of the pre-trained model. To address this challenge, we further introduce and formalize a causal debiasing perspective, which is substantiated via a hierarchical Bayesian deep learning model, named PreRec. Our empirical studies on real-world data show that the proposed model could significantly improve the recommendation performance in zero- and few-shot learning settings under both cross-market and cross-platform scenarios. | cs | 0 |
Title: A note on a flip-connected class of generalized domino tilings of the box $[0,2]^n$
Abstract: Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an $(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two $(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and $\{C,D\}\neq \{A,B\}$, then the pair $C,D$ is called a flip of $A,B$. A family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^n$ if interiors of every two members of $\mathscr{D}$ are disjoint and $\bigcup_{B\in \mathscr{D}}B=[0,2]^n$. An $(n-d)$-domino tiling $\mathscr{D}'$ is obtained from an $(n-d)$-domino tiling $\mathscr{D}$ by a flip, if there is a twin pair $A,B\in \mathscr{D}$ such that $\mathscr{D}'=(\mathscr{D}\setminus \{A,B\})\cup \{C,D\}$, where $C,D$ is a flip of $A,B$. A family of $(n-d)$-domino tilings of the box $[0,2]^n$ is flip-connected, if for every two members $\mathscr{D},\mathscr{E}$ of this family the tiling $\mathscr{E}$ can be obtained from $\mathscr{D}$ by a sequence of flips. In the paper some flip-connected class of $(n-d)$-domino tilings of the box $[0,2]^n$ is described. | math | 0 |
Title: Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games
Abstract: We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are beyond reach with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems. | cs | 1 |
Title: Projections, Embeddings and Stability
Abstract: In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and practical implications; the theory applies even if the boundary operator is rank deficient, or near rank deficient. If desired, the pseudoinverse can be implemented directly using standard tools like Matlab. We also introduce a new and simplified version of the semidiscrete approximation of the linear PDE system, which completely avoids taking the time derivative of the boundary data. The stability results are valid for general, nondiagonal summation-by-parts norms. Another key result is the extension of summation-by-parts operators to multi-domains by means of carefully crafted embedding operators. No extra numerical boundary conditions are required at the grid interfaces. The aforementioned pseudoinverse allows for a compact representation of these multi-block operators, which preserves all relevant properties of the single-block operators. The embedding operators can be constructed for multiple space dimensions. Numerical results for the two-dimensional Maxwell's equations are presented, and they show very good agreement with theory. | math | 0 |
Title: Set-valued prediction in hierarchical classification with constrained representation complexity
Abstract: Set-valued prediction is a well-known concept in multi-class classification. When a classifier is uncertain about the class label for a test instance, it can predict a set of classes instead of a single class. In this paper, we focus on hierarchical multi-class classification problems, where valid sets (typically) correspond to internal nodes of the hierarchy. We argue that this is a very strong restriction, and we propose a relaxation by introducing the notion of representation complexity for a predicted set. In combination with probabilistic classifiers, this leads to a challenging inference problem for which specific combinatorial optimization algorithms are needed. We propose three methods and evaluate them on benchmark datasets: a na\"ive approach that is based on matrix-vector multiplication, a reformulation as a knapsack problem with conflict graph, and a recursive tree search method. Experimental results demonstrate that the last method is computationally more efficient than the other two approaches, due to a hierarchical factorization of the conditional class distribution. | cs | 1 |
Title: Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator
Abstract: We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lov{\'a}sz--Schrijver SDP operator $\LS_+$, with a particular focus on a search for relatively small graphs with high $\LS_+$-rank (the least number of iterations of the $\LS_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\LS_+$-rank is asymptotically a linear function of its number of vertices, which is the best possible up to improvements in the constant factor (previous best result in this direction, from 1999, yielded graphs whose $\LS_+$-rank only grew with the square root of the number of vertices). | math | 0 |
Title: Quotients of the braid group that are extensions of the symmetric group
Abstract: We consider normal subgroups $N$ of the braid group $B_n$ such that the quotient $B_n/N$ is an extension of the symmetric group by an abelian group. We show that, if $n\geq 4$, then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions $(n-1,1)$ and $(n-2,2)$, working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions. | math | 0 |
Title: Real-and-Present: Investigating the Use of Life-Size 2D Video Avatars in HMD-Based AR Teleconferencing
Abstract: Augmented Reality (AR) teleconferencing allows separately located users to interact with each other in 3D through agents in their own physical environments. Existing methods leveraging volumetric capturing and reconstruction can provide a high-fidelity experience but are often too complex and expensive for everyday usage. Other solutions target mobile and effortless-to-setup teleconferencing on AR Head Mounted Displays (HMD). They directly transplant the conventional video conferencing onto an AR-HMD platform or use avatars to represent remote participants. However, they can only support either a high fidelity or a high level of co-presence. Moreover, the limited Field of View (FoV) of HMDs could further influence users' immersive experience. To achieve a balance between fidelity and co-presence, we explore using life-size 2D video-based avatars (video avatars for short) in AR teleconferencing. Specifically, with the potential effect of FoV on users' perception of proximity, we first conduct a pilot study to explore the local-user-centered optimal placement of video avatars in small-group AR conversations. With the placement results, we then implement a proof-of-concept prototype of video-avatar-based teleconferencing. We conduct user evaluations with the prototype to verify its effectiveness in balancing fidelity and co-presence. Following the indication in the pilot study, we further quantitatively explore the effect of FoV size on the video avatar's optimal placement through a user study involving more FoV conditions in a VR-simulated environment. We regress placement models to serve as references for computationally determining video avatar placements in such teleconferencing applications on various existing AR HMDs and future ones with bigger FoVs. | cs | 0 |
Title: Brown representability for directed graphs
Abstract: We prove that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is representable. | math | 1 |
Title: An Achievable Rate-Distortion Region for the Multiple Descriptions Problem
Abstract: A multiple-descriptions (MD) coding strategy is proposed and an inner bound to the achievable rate-distortion region is derived. The scheme utilizes linear codes. It is shown in two different MD set-ups that the linear coding scheme achieves a larger rate-distortion region than previously known random coding strategies. Furthermore, it is shown via an example that the best known random coding scheme for the set-up can be improved by including additional randomly generated codebooks. | cs | 1 |
Title: Will 6G be Semantic Communications? Opportunities and Challenges from Task Oriented and Secure Communications to Integrated Sensing
Abstract: This paper explores opportunities and challenges of task (goal)-oriented and semantic communications for next-generation (NextG) communication networks through the integration of multi-task learning. This approach employs deep neural networks representing a dedicated encoder at the transmitter and multiple task-specific decoders at the receiver, collectively trained to handle diverse tasks including semantic information preservation, source input reconstruction, and integrated sensing and communications. To extend the applicability from point-to-point links to multi-receiver settings, we envision the deployment of decoders at various receivers, where decentralized learning addresses the challenges of communication load and privacy concerns, leveraging federated learning techniques that distribute model updates across decentralized nodes. However, the efficacy of this approach is contingent on the robustness of the employed deep learning models. We scrutinize potential vulnerabilities stemming from adversarial attacks during both training and testing phases. These attacks aim to manipulate both the inputs at the encoder at the transmitter and the signals received over the air on the receiver side, highlighting the importance of fortifying semantic communications against potential multi-domain exploits. Overall, the joint and robust design of task-oriented communications, semantic communications, and integrated sensing and communications in a multi-task learning framework emerges as the key enabler for context-aware, resource-efficient, and secure communications ultimately needed in NextG network systems. | cs | 0 |
Title: Explicit separations between randomized and deterministic Number-on-Forehead communication
Abstract: We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function $f:[N]^3 \to \{0,1\}$, such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing it requires sending about $(\log N)^{1/3}$ many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result of the first and third authors on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting. | cs | 0 |
Title: Joint symbolic aggregate approximation of time series
Abstract: The increasing availability of temporal data poses a challenge to time-series and signal-processing domains due to its high numerosity and complexity. Symbolic representation outperforms raw data in a variety of engineering applications due to its storage efficiency, reduced numerosity, and noise reduction. The most recent symbolic aggregate approximation technique called ABBA demonstrates outstanding performance in preserving essential shape information of time series and enhancing the downstream applications. However, ABBA cannot handle multiple time series with consistent symbols, i.e., the same symbols from distinct time series are not identical. Also, working with appropriate ABBA digitization involves the tedious task of tuning the hyperparameters, such as the number of symbols or tolerance. Therefore, we present a joint symbolic aggregate approximation that has symbolic consistency, and show how the hyperparameter of digitization can itself be optimized alongside the compression tolerance ahead of time. Besides, we propose a novel computing paradigm that enables parallel computing of symbolic approximation. The extensive experiments demonstrate its superb performance and outstanding speed regarding symbolic approximation and reconstruction. | cs | 0 |
Title: Development of An Autonomous Bridge Deck Inspection Robotic System
Abstract: The threat to safety of aging bridges has been recognized as a critical concern to the general public due to the poor condition of many bridges in the U.S. Currently, the bridge inspection is conducted manually, and it is not efficient to identify bridge condition deterioration in order to facilitate implementation of appropriate maintenance or rehabilitation procedures. In this paper, we report a new development of the autonomous mobile robotic system for bridge deck inspection and evaluation. The robot is integrated with several nondestructive evaluation (NDE) sensors and a navigation control algorithm to allow it to accurately and autonomously maneuver on the bridge deck to collect visual images and conduct NDE measurements. The developed robotic system can reduce the cost and time of the bridge deck data collection and inspection. For efficient bridge deck monitoring, the crack detection algorithm to build the deck crack map is presented in detail. The impact-echo (IE), ultrasonic surface waves (USW) and electrical resistivity (ER) data collected by the robot are analyzed to generate the delamination, concrete elastic modulus, corrosion maps of the bridge deck, respectively. The presented robotic system has been successfully deployed to inspect numerous bridges in more than ten different states in the U.S. | cs | 1 |
Title: Sharing Non-Anonymous Costs of Multiple Resources Optimally
Abstract: In cost sharing games, the existence and efficiency of pure Nash equilibria fundamentally depends on the method that is used to share the resources' costs. We consider a general class of resource allocation problems in which a set of resources is used by a heterogeneous set of selfish users. The cost of a resource is a (non-decreasing) function of the set of its users. Under the assumption that the costs of the resources are shared by uniform cost sharing protocols, i.e., protocols that use only local information of the resource's cost structure and its users to determine the cost shares, we exactly quantify the inefficiency of the resulting pure Nash equilibria. Specifically, we show tight bounds on prices of stability and anarchy for games with only submodular and only supermodular cost functions, respectively, and an asymptotically tight bound for games with arbitrary set-functions. While all our upper bounds are attained for the well-known Shapley cost sharing protocol, our lower bounds hold for arbitrary uniform cost sharing protocols and are even valid for games with anonymous costs, i.e., games in which the cost of each resource only depends on the cardinality of the set of its users. | cs | 1 |
Title: No solitary waves in 2-d gravity and capillary waves in deep water
Abstract: A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Precisely, we consider incompressible, irrotational, infinite depth water wave equation, either with gravity and without surface tension, or without gravity but with surface tension. In both of these cases we prove that there are no solitary waves. | math | 1 |
Title: Efficient Private SCO for Heavy-Tailed Data via Clipping
Abstract: We consider stochastic convex optimization for heavy-tailed data with the guarantee of being differentially private (DP). Prior work on this problem is restricted to the gradient descent (GD) method, which is inefficient for large-scale problems. In this paper, we resolve this issue and derive the first high-probability bounds for the private stochastic method with clipping. For general convex problems, we derive excess population risks $\Tilde{O}\left(\frac{d^{1/7}\sqrt{\ln\frac{(n \epsilon)^2}{\beta d}}}{(n\epsilon)^{2/7}}\right)$ and $\Tilde{O}\left(\frac{d^{1/7}\ln\frac{(n\epsilon)^2}{\beta d}}{(n\epsilon)^{2/7}}\right)$ under bounded or unbounded domain assumption, respectively (here $n$ is the sample size, $d$ is the dimension of the data, $\beta$ is the confidence level and $\epsilon$ is the private level). Then, we extend our analysis to the strongly convex case and non-smooth case (which works for generalized smooth objectives with H$\ddot{\text{o}}$lder-continuous gradients). We establish new excess risk bounds without bounded domain assumption. The results above achieve lower excess risks and gradient complexities than existing methods in their corresponding cases. Numerical experiments are conducted to justify the theoretical improvement. | cs | 1 |
Title: Core equality of real sequences
Abstract: Given an ideal $\mathcal{I}$ on $\omega$ and a bounded real sequence $\textbf{x}$, we denote by $\text{core}_{\textbf{x}}(\mathcal{I})$ the smallest interval $[a,b]$ such that $\{n \in \omega: x_n \notin [a-\varepsilon,b+\varepsilon]\} \in \mathcal{I}$ for all $\varepsilon>0$ (which corresponds to the interval $[\,\liminf \textbf{x}, \limsup \textbf{x}\,]$ if $\mathcal{I}$ is the ideal $\text{Fin}$ of finite subsets of $\omega$). First, we characterize all the infinite real matrices $A$ such that $$ \text{core}_{A\textbf{x}}(\mathcal{J})=\text{core}_{\textbf{x}}(\mathcal{I}) $$ for all bounded sequences $\textbf{x}$, provided that $\mathcal{J}$ is a countably generated ideal on $\omega$ and $A$ maps bounded sequences into bounded sequences. Such characterization fails if both $\mathcal{I}$ and $\mathcal{J}$ are the ideal of asymptotic density zero sets. Next, we show that such equality is possible for distinct ideals $\mathcal{I}, \mathcal{J}$, answering an open question in [J.~Math.~Anal.~Appl.~\textbf{321} (2006), 515--523]. Lastly, we prove that, if $\mathcal{J}=\text{Fin}$, the above equality holds for some matrix $A$ if and only if $\mathcal{I}=\text{Fin}$ or $\mathcal{I}=\text{Fin}\oplus \mathcal{P}(\omega)$. | math | 0 |
Title: Hybrid Learning of Time-Series Inverse Dynamics Models for Locally Isotropic Robot Motion
Abstract: Applications of force control and motion planning often rely on an inverse dynamics model to represent the high-dimensional dynamic behavior of robots during motion. The widespread occurrence of low-velocity, small-scale, locally isotropic motion (LIMO) typically complicates the identification of appropriate models due to the exaggeration of dynamic effects and sensory perturbation caused by complex friction and phenomena of hysteresis, e.g., pertaining to joint elasticity. We propose a hybrid model learning base architecture combining a rigid body dynamics model identified by parametric regression and time-series neural network architectures based on multilayer-perceptron, LSTM, and Transformer topologies. Further, we introduce novel joint-wise rotational history encoding, reinforcing temporal information to effectively model dynamic hysteresis. The models are evaluated on a KUKA iiwa 14 during algorithmically generated locally isotropic movements. Together with the rotational encoding, the proposed architectures outperform state-of-the-art baselines by a magnitude of 10$^3$ yielding an RMSE of 0.14 Nm. Leveraging the hybrid structure and time-series encoding capabilities, our approach allows for accurate torque estimation, indicating its applicability in critically force-sensitive applications during motion sequences exceeding the capacity of conventional inverse dynamics models while retaining trainability in face of scarce data and explainability due to the employed physics model prior. | cs | 1 |
Title: Solving Almost all Systems of Random Quadratic Equations
Abstract: This paper deals with finding an $n$-dimensional solution $x$ to a system of quadratic equations of the form $y_i=|\langle{a}_i,x\rangle|^2$ for $1\le i \le m$, which is also known as phase retrieval and is NP-hard in general. We put forth a novel procedure for minimizing the amplitude-based least-squares empirical loss, that starts with a weighted maximal correlation initialization obtainable with a few power or Lanczos iterations, followed by successive refinements based upon a sequence of iteratively reweighted (generalized) gradient iterations. The two (both the initialization and gradient flow) stages distinguish themselves from prior contributions by the inclusion of a fresh (re)weighting regularization technique. The overall algorithm is conceptually simple, numerically scalable, and easy-to-implement. For certain random measurement models, the novel procedure is shown capable of finding the true solution $x$ in time proportional to reading the data $\{(a_i;y_i)\}_{1\le i \le m}$. This holds with high probability and without extra assumption on the signal $x$ to be recovered, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns in the signal vector, namely, $m>cn$. Empirically, the upshots of this contribution are: i) (almost) $100\%$ perfect signal recovery in the high-dimensional (say e.g., $n\ge 2,000$) regime given only an information-theoretic limit number of noiseless equations, namely, $m=2n-1$ in the real-valued Gaussian case; and, ii) (nearly) optimal statistical accuracy in the presence of additive noise of bounded support. Finally, substantial numerical tests using both synthetic data and real images corroborate markedly improved signal recovery performance and computational efficiency of our novel procedure relative to state-of-the-art approaches. | math | 1 |
Title: Joint Multi-Facts Reasoning Network For Complex Temporal Question Answering Over Knowledge Graph
Abstract: Temporal Knowledge Graph (TKG) is an extension of regular knowledge graph by attaching the time scope. Existing temporal knowledge graph question answering (TKGQA) models solely approach simple questions, owing to the prior assumption that each question only contains a single temporal fact with explicit/implicit temporal constraints. Hence, they perform poorly on questions which own multiple temporal facts. In this paper, we propose \textbf{\underline{J}}oint \textbf{\underline{M}}ulti \textbf{\underline{F}}acts \textbf{\underline{R}}easoning \textbf{\underline{N}}etwork (JMFRN), to jointly reasoning multiple temporal facts for accurately answering \emph{complex} temporal questions. Specifically, JMFRN first retrieves question-related temporal facts from TKG for each entity of the given complex question. For joint reasoning, we design two different attention (\ie entity-aware and time-aware) modules, which are suitable for universal settings, to aggregate entities and timestamps information of retrieved facts. Moreover, to filter incorrect type answers, we introduce an additional answer type discrimination task. Extensive experiments demonstrate our proposed method significantly outperforms the state-of-art on the well-known complex temporal question benchmark TimeQuestions. | cs | 0 |
Title: Probabilistic Modeling for Sequences of Sets in Continuous-Time
Abstract: Neural marked temporal point processes have been a valuable addition to the existing toolbox of statistical parametric models for continuous-time event data. These models are useful for sequences where each event is associated with a single item (a single type of event or a "mark") -- but such models are not suited for the practical situation where each event is associated with a set of items. In this work, we develop a general framework for modeling set-valued data in continuous-time, compatible with any intensity-based recurrent neural point process model. In addition, we develop inference methods that can use such models to answer probabilistic queries such as "the probability of item $A$ being observed before item $B$," conditioned on sequence history. Computing exact answers for such queries is generally intractable for neural models due to both the continuous-time nature of the problem setting and the combinatorially-large space of potential outcomes for each event. To address this, we develop a class of importance sampling methods for querying with set-based sequences and demonstrate orders-of-magnitude improvements in efficiency over direct sampling via systematic experiments with four real-world datasets. We also illustrate how to use this framework to perform model selection using likelihoods that do not involve one-step-ahead prediction. | cs | 0 |
Title: New Generalizations of the Bethe Approximation via Asymptotic Expansion
Abstract: The Bethe approximation, discovered in statistical physics, gives an efficient algorithm called belief propagation (BP) for approximating a partition function. BP empirically gives an accurate approximation for many problems, e.g., low-density parity-check codes, compressed sensing, etc. Recently, Vontobel gives a novel characterization of the Bethe approximation using graph cover. In this paper, a new approximation based on the Bethe approximation is proposed. The new approximation is derived from Vontobel's characterization using graph cover, and expressed by using the edge zeta function, which is related with the Hessian of the Bethe free energy as shown by Watanabe and Fukumizu. On some conditions, it is proved that the new approximation is asymptotically better than the Bethe approximation. | cs | 1 |
Title: Large Language Models Relearn Removed Concepts
Abstract: Advances in model editing through neuron pruning hold promise for removing undesirable concepts from large language models. However, it remains unclear whether models have the capacity to reacquire pruned concepts after editing. To investigate this, we evaluate concept relearning in models by tracking concept saliency and similarity in pruned neurons during retraining. Our findings reveal that models can quickly regain performance post-pruning by relocating advanced concepts to earlier layers and reallocating pruned concepts to primed neurons with similar semantics. This demonstrates that models exhibit polysemantic capacities and can blend old and new concepts in individual neurons. While neuron pruning provides interpretability into model concepts, our results highlight the challenges of permanent concept removal for improved model \textit{safety}. Monitoring concept reemergence and developing techniques to mitigate relearning of unsafe concepts will be important directions for more robust model editing. Overall, our work strongly demonstrates the resilience and fluidity of concept representations in LLMs post concept removal. | cs | 0 |
Title: From Spot 2.0 to Spot 2.10: What's New?
Abstract: Spot is a C ++ 17 library for LTL and $\omega$-automata manipulation, with command-line utilities, and Python bindings. This paper summarizes its evolution over the past six years, since the release of Spot 2.0, which was the first version to support $\omega$-automata with arbitrary acceptance conditions, and the last version presented at a conference. Since then, Spot has been extended with several features such as acceptance transformations, alternating automata, games, LTL synthesis, and more. We also shed some lights on the data-structure used to store automata. | cs | 1 |
Title: Homological properties of the relative Frobenius morphism
Abstract: This work concerns maps of commutative noetherian local rings containing a field of positive characteristic. Given such a map $\varphi$ of finite flat dimension, the results relate homological properties of the relative Frobenius of $\varphi$ to those of the fibers of $\varphi$. The focus is on the complete intersection property and the Gorenstein property. | math | 0 |
Title: Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system
Abstract: We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions. | math | 0 |
Title: Distribution of primes represented by polynomials and Multiple Dedekind zeta functions
Abstract: n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadratic fields. The conjecture for pairs of primes and multiple Dedekind zeta function over the Gaussian integers provide error less than a tenth of a percent. We also tested conjectures that compare sets of primes in a pair of different quadratic fields. Numerically, such quotients can be expressed in terms of regulators and class numbers. Some of the data, together with the code, is available on GitHub, (see \cite{Zouberou}). | math | 1 |
Title: Rellich inequalities in bounded domains
Abstract: We find necessary and sufficient conditions for the validity of weighted Rellich inequalities in Lp for functions in bounded domains vanishing at the boundary. General operators like L = Delta+ c\|x|^2x nabla-b\|x|^2 are considered. Critical cases and remainder terms are also investigated. | math | 1 |
Title: Enhancing RAW-to-sRGB with Decoupled Style Structure in Fourier Domain
Abstract: RAW to sRGB mapping, which aims to convert RAW images from smartphones into RGB form equivalent to that of Digital Single-Lens Reflex (DSLR) cameras, has become an important area of research. However, current methods often ignore the difference between cell phone RAW images and DSLR camera RGB images, a difference that goes beyond the color matrix and extends to spatial structure due to resolution variations. Recent methods directly rebuild color mapping and spatial structure via shared deep representation, limiting optimal performance. Inspired by Image Signal Processing (ISP) pipeline, which distinguishes image restoration and enhancement, we present a novel Neural ISP framework, named FourierISP. This approach breaks the image down into style and structure within the frequency domain, allowing for independent optimization. FourierISP is comprised of three subnetworks: Phase Enhance Subnet for structural refinement, Amplitude Refine Subnet for color learning, and Color Adaptation Subnet for blending them in a smooth manner. This approach sharpens both color and structure, and extensive evaluations across varied datasets confirm that our approach realizes state-of-the-art results. Code will be available at ~\url{https://github.com/alexhe101/FourierISP}. | cs | 0 |
Title: Characterization of commuting graphs of finite groups having small genus
Abstract: In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only $K_8 \sqcup 9K_1$, $K_8 \sqcup 5K_2$, $K_8 \sqcup 3K_4$, $K_8 \sqcup 9K_3$, $K_8\sqcup 9(K_1 \vee 3K_2)$, $3K_6$ and $3K_6 \sqcup 4K_4 \sqcup 6K_2$ can be realized as commuting graphs of finite groups. As consequences of our results we also show that for any finite non-abelian group $G$ if the commuting graph of $G$ (denoted by $\Gamma_c(G)$) is double-toroidal or triple-toroidal then $\Gamma_c(G)$ and its complement satisfy Hansen-Vuki{\v{c}}evi{\'c} Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group $(\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8$, that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs. | math | 0 |
Title: Contravariant Pseudo-Hessian manifolds and their associated Poisson structures
Abstract: A contravariant pseudo-Hessian manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a contravariant Codazzi equation. When $h$ is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties similar to Poisson manifolds and, in fact, to any contravariant pseudo-Hessian manifold $(M,\nabla,h)$ we associate naturally a Poisson tensor on $TM$. We investigate these properties and we study in details many classes of such structures in order to highlight the richness of the geometry of these manifolds. | math | 1 |
Title: On the conductor of cohomological transforms
Abstract: In the analytic study of trace functions of $\ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such a transform is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project. | math | 1 |
Title: Expressive Speech-driven Facial Animation with controllable emotions
Abstract: It is in high demand to generate facial animation with high realism, but it remains a challenging task. Existing approaches of speech-driven facial animation can produce satisfactory mouth movement and lip synchronization, but show weakness in dramatic emotional expressions and flexibility in emotion control. This paper presents a novel deep learning-based approach for expressive facial animation generation from speech that can exhibit wide-spectrum facial expressions with controllable emotion type and intensity. We propose an emotion controller module to learn the relationship between the emotion variations (e.g., types and intensity) and the corresponding facial expression parameters. It enables emotion-controllable facial animation, where the target expression can be continuously adjusted as desired. The qualitative and quantitative evaluations show that the animation generated by our method is rich in facial emotional expressiveness while retaining accurate lip movement, outperforming other state-of-the-art methods. | cs | 0 |
Title: $p$-numerical semigroups of Pell triples
Abstract: For a nonnegative integer $p$, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots,a_\kappa$ with $\gcd(a_1,a_2,\dots,a_\kappa)=1$ are expressed in more than $p$ ways. When $p=0$, $S=S_0$ is the original numerical semigroup. The largest element and the cardinality of $\mathbb N_0\backslash S_p$ are called the $p$-Frobenius number and the $p$-genus, respectively. Their explicit formulas are known for $\kappa=2$, but those for $\kappa\ge 3$ have been found only in some special cases. For some known cases, such as the Fibonacci and the Jacobsthal triplets, similar techniques could be applied and explicit formulas such as the $p$-Frobenius number could be found. In this paper, we give explicit formulas for the $p$-Frobenius number and the $p$-genus of Pell numerical semigroups $\bigl(P_i(u),P_{i+2}(u),P_{i+k}(u)\bigr)$. Here, for a given positive integer $u$, Pell-type numbers $P_n(u)$ satisfy the recurrence relation $P_n(u)=u P_{n-1}(u)+P_{n-2}(u)$ ($n\ge 2$) with $P_0(u)=0$ and $P_1(u)=1$. The $p$-Ap\'ery set is used to find the formulas, but it shows a different pattern from those in the known results, and some case by case discussions are necessary. | math | 0 |
Title: Generalised Local Fractional Hermite-Hadamard Type Inequalities on Fractal Sets
Abstract: Fractal geometry and analysis constitute a growing field, with numerous applications, based on the principles of fractional calculus. Fractals sets are highly effective in improving convex inequalities and their generalisations. In this paper, we establish a generalized notion of convexity. By defining generalised $\phi_{h-s}$ convex functions, we extend the well known concepts of generalised convex functions, $P$-functions, Breckner $s$-convex functions, $h$-convex functions amongst others. With this definition, we prove Hermite-Hadamard type inequalities for generalized $\phi_{h-s}$ convex mappings onto fractal sets. Our results are then applied to probability theory. | math | 0 |
Title: The Slepian model based independent interval approximation of persistency and zero-level exceedance distributions
Abstract: In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used. | math | 0 |
Title: Vanishing Cycles in Holomorphic Foliations by Curves and Foliated Shells
Abstract: The purpose of this paper is the study of vanishing cycles in holomorphic foliations by complex curves on compact complex manifolds. The main result consists in showing that a vanishing cycle comes together with a much richer complex geometric object - we call this object a foliated shell. | math | 1 |
Title: Sharper Bounds for $\ell_p$ Sensitivity Sampling
Abstract: In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension $d$ and the total sensitivity $\mathfrak S$ in remarkably general settings. However, guarantees going beyond this general bound of $\mathfrak S d$ are known in perhaps only one setting, for $\ell_2$ subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for $\ell_p$ subspace embeddings for $p > 2$ that improve over the general $\mathfrak S d$ bound, achieving a bound of roughly $\mathfrak S^{2-2/p}$ for $2<p<\infty$. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly $d$ for $1\leq p<2$, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly $d^{2/p}\mathfrak S^{2-4/p}$ for $2<p<\infty$. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small $\ell_p$ sensitivity. | cs | 0 |
Title: Twinless articulation points and some related problems
Abstract: Let $G=(V,E)$ be a twinless strongly connected graph. a vertex $v\in V$ is a twinless articulation point if the subrgraph obtained from $G$ by removing the vertex $v$ is not twinless strongly connected. An edge $e\in E$ is a twinless bridge if the subgraph obtained from $G$ by deleting $e$ is not twiless strongly connected graph. In this paper we study twinless articulation points and twinless bridges. We also study the problem of finding a minimum cardinality edge subset $E_{1} \subseteq E$ such that the subgraph $(V,E_{1})$ is twinless strongly connected. Moreover, we present an algorithm for computing the $2$-vertex-twinless connected components of $G$. | cs | 1 |
Title: The Wave Equation on Lattices and Oscillatory Integrals
Abstract: In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove that the optimal time decay rate of the fundamental solution is of order $|t|^{-\frac{3}{2}}\log |t|$, which is the first extension of P. Schultz's \cite{S98} results in $d=2,3$ to the higher dimension. We also observe that the decay rate for $d=2,3,4$ can be well interpreted by the Newton polyhedra. Moreover, we prove $l^p\rightarrow l^q$ estimates as well as Strichartz estimates and give applications to nonlinear wave equations. | math | 0 |
Title: Magnitude function identifies generic finite metric spaces
Abstract: We show that a ``generic'' finite metric space can be identified by the asymptotic behavior of the magnitude function. In particular, almost every finite set in Euclidean space can be determined by the magnitude function. | math | 0 |
Title: Source-Free Online Domain Adaptive Semantic Segmentation of Satellite Images under Image Degradation
Abstract: Online adaptation to distribution shifts in satellite image segmentation stands as a crucial yet underexplored problem. In this paper, we address source-free and online domain adaptation, i.e., test-time adaptation (TTA), for satellite images, with the focus on mitigating distribution shifts caused by various forms of image degradation. Towards achieving this goal, we propose a novel TTA approach involving two effective strategies. First, we progressively estimate the global Batch Normalization (BN) statistics of the target distribution with incoming data stream. Leveraging these statistics during inference has the ability to effectively reduce domain gap. Furthermore, we enhance prediction quality by refining the predicted masks using global class centers. Both strategies employ dynamic momentum for fast and stable convergence. Notably, our method is backpropagation-free and hence fast and lightweight, making it highly suitable for on-the-fly adaptation to new domain. Through comprehensive experiments across various domain adaptation scenarios, we demonstrate the robust performance of our method. | cs | 0 |
Title: More on Arago'n Artacho - Campoy's Algorithm
Abstract: The Arago'n Artacho-Campoy algorithm (AACA) is a new method for finding zeros of sums of monotone operators. In this paper we complete the analysis of [2] and[1] by providing study of the two possible Arago'n Artacho-Campoy operators. | math | 0 |
Title: A taxonomy of estimator consistency on discrete estimation problems
Abstract: We describe a four-level hierarchy mapping both all discrete estimation problems and all estimators on these problems, such that the hierarchy describes each estimator's consistency guarantees on each problem class. We show that no estimator is consistent for all estimation problems, but that some estimators, such as Maximum A Posteriori, are consistent for the widest possible class of discrete estimation problems. For Maximum Likelihood and Approximate Maximum Likelihood estimators we show that they do not provide consistency on as wide a class, but define a sub-class of problems characterised by their consistency. Lastly, we show that some popular estimators, specifically Strict Minimum Message Length, do not provide consistency guarantees even within the sub-class. | math | 1 |
Title: On the number of hyperbolic Dehn fillings of a given volume
Abstract: Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is quadratic. We show that there exists c=c(M) such that the number of hyperbolic Dehn fillings of M with any given volume v is uniformly bounded by c. | math | 1 |
Title: Transformer Neural Autoregressive Flows
Abstract: Density estimation, a central problem in machine learning, can be performed using Normalizing Flows (NFs). NFs comprise a sequence of invertible transformations, that turn a complex target distribution into a simple one, by exploiting the change of variables theorem. Neural Autoregressive Flows (NAFs) and Block Neural Autoregressive Flows (B-NAFs) are arguably the most perfomant members of the NF family. However, they suffer scalability issues and training instability due to the constraints imposed on the network structure. In this paper, we propose a novel solution to these challenges by exploiting transformers to define a new class of neural flows called Transformer Neural Autoregressive Flows (T-NAFs). T-NAFs treat each dimension of a random variable as a separate input token, using attention masking to enforce an autoregressive constraint. We take an amortization-inspired approach where the transformer outputs the parameters of an invertible transformation. The experimental results demonstrate that T-NAFs consistently match or outperform NAFs and B-NAFs across multiple datasets from the UCI benchmark. Remarkably, T-NAFs achieve these results using an order of magnitude fewer parameters than previous approaches, without composing multiple flows. | cs | 0 |
Title: On the convergence of stochastic transport equations to a deterministic parabolic one
Abstract: A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension $d\geq 2$; the case $d=1$ is also investigated but no conclusive answer is obtained. | math | 1 |
Title: Concentration of maximum degree in random planar graphs
Abstract: Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\leq 1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values. In contrast, this is not true anymore in the dense regime, when $m/n>1$, where the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with bounded size. | math | 1 |
Title: Effective descent morphisms of filtered preorders
Abstract: We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder $X$, is defined as a preorder $A$ equipped with a profunctor $X\to A$ and, equivalently, as a set $A$ equipped with a family $(A_x)_{x\in X}$ of upclosed subsets of $A$ with $x'\leqslant x\Rightarrow A_x\subseteq A_{x'}$. | math | 0 |
Title: On a new method for the stochastic perturbation of the disease transmission coefficient in SIS Models
Abstract: In this study we investigate a novel approach to stochastically perturb the disease transmission coefficient, which is a key parameter in susceptible-infected-susceptible (SIS) models. Motivated by the papers [2] and [5], we perturb the disease transmission coefficient with a Gaussian white noise, formally modelled as the time derivative of a mean reverting Ornstein-Uhlenbeck process. We remark that, thanks to a suitable representation of the solution to the deterministic SIS model, this perturbation is rigorous and supported by a Wong-Zakai approximation argument that consists in smoothing the singular Gaussian white noise and then taking limit of the solution from the approximated model. We prove that the stochastic version of the classic SIS model obtained this way preserves a crucial feature of the deterministic equation: the reproduction number dictating the two possible asymptotic regimes for the infection, i.e. extinction and persistence, remains unchanged. We then identify the class of perturbing noises for which this property holds and propose simple sufficient conditions for that. All the theoretical discoveries are illustrated and discussed with the help of several numerical simulations. | math | 1 |