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Title: Probabilistic Trajectory Design Via Approximate Gaussian Mixture Steering
Abstract: A method is presented to solve a stochastic, nonlinear optimal control problem representative of spacecraft trajectory design under uncertainty. The problem is reformulated as a chance constrained nonlinear program, or what is known as a distribution steering problem. Typical distribution steering problems rely on the underlying uncertainties to be Gaussian distributions. This work expands on previous developments by embedding Gaussian mixture distributions into the formulation to better handle the uncertainty propagation and chance constraints involved. The method is applied to a finite-thrust Earth-to-Mars transfer problem. Evaluation via Monte Carlo analysis shows a greater satisfaction of constraints under non-Gaussian distributions of the state and a statistically lower cost. | math | 0 |
Title: A General Implicit Framework for Fast NeRF Composition and Rendering
Abstract: A variety of Neural Radiance Fields (NeRF) methods have recently achieved remarkable success in high render speed. However, current accelerating methods are specialized and incompatible with various implicit methods, preventing real-time composition over various types of NeRF works. Because NeRF relies on sampling along rays, it is possible to provide general guidance for acceleration. To that end, we propose a general implicit pipeline for composing NeRF objects quickly. Our method enables the casting of dynamic shadows within or between objects using analytical light sources while allowing multiple NeRF objects to be seamlessly placed and rendered together with any arbitrary rigid transformations. Mainly, our work introduces a new surface representation known as Neural Depth Fields (NeDF) that quickly determines the spatial relationship between objects by allowing direct intersection computation between rays and implicit surfaces. It leverages an intersection neural network to query NeRF for acceleration instead of depending on an explicit spatial structure.Our proposed method is the first to enable both the progressive and interactive composition of NeRF objects. Additionally, it also serves as a previewing plugin for a range of existing NeRF works. | cs | 0 |
Title: Calderon-Zygmund theory for strongly coupled linear system of nonlocal equations with Holder-regular coefficient
Abstract: We extend the Calder\'on-Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations $\mathcal{L}^{s}_{A} u = f$, where the operator $\mathcal{L}^{s}_{A}$ is formally given by \[ \mathcal{L}^s_{A}u = \int_{\mathbb{R}^n}\frac{A(x, y)}{\vert x-y\vert ^{n+2s}} \frac{(x-y)\otimes (x-y)}{\vert x-y\vert ^2}(u(x)-u(y))dy. \] For $0 < s < 1$ and $A:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}$ taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier-Lam\'e linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot, y)$ is uniformly Holder continuous and $\inf_{x\in \mathbb{R}^n}A(x, x) > 0$, then for $f\in L^{p}_{loc},$ for $p\geq 2$, the solution vector $u\in H^{2s-\delta,p}_{loc}$ for some $\delta\in (0, s)$. | math | 0 |
Title: Tangent categories of algebras over operads
Abstract: Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper. | math | 1 |
Title: Hilbert space operators with compatible off-diagonal corners
Abstract: Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\| P T (I-P) \| = \| (I-P) T P \|$ for all orthogonal projections $P$ on $\mathcal{H}$. When $\mathcal{H}$ is finite-dimensional, we also obtain a complete characterization of those operators for which $\mathrm{rank}\, (I-P) T P = \mathrm{rank}\, P T (I-P)$ for all orthogonal projections $P$. When $\mathcal{H}$ is infinite-dimensional, we show that any operator with the latter property is normal, and its spectrum is contained in either a line or a circle in the complex plane. | math | 1 |
Title: Anchor Pruning for Object Detection
Abstract: This paper proposes anchor pruning for object detection in one-stage anchor-based detectors. While pruning techniques are widely used to reduce the computational cost of convolutional neural networks, they tend to focus on optimizing the backbone networks where often most computations are. In this work we demonstrate an additional pruning technique, specifically for object detection: anchor pruning. With more efficient backbone networks and a growing trend of deploying object detectors on embedded systems where post-processing steps such as non-maximum suppression can be a bottleneck, the impact of the anchors used in the detection head is becoming increasingly more important. In this work, we show that many anchors in the object detection head can be removed without any loss in accuracy. With additional retraining, anchor pruning can even lead to improved accuracy. Extensive experiments on SSD and MS COCO show that the detection head can be made up to 44% more efficient while simultaneously increasing accuracy. Further experiments on RetinaNet and PASCAL VOC show the general effectiveness of our approach. We also introduce `overanchorized' models that can be used together with anchor pruning to eliminate hyperparameters related to the initial shape of anchors. Code and models are available at https://github.com/Mxbonn/anchor_pruning. | cs | 1 |
Title: Primitive elements in infinitesimal bialgebras
Abstract: For any set S, the free magmatic algebra spanned by card(S) binary products is the vector space spanned by the set of all planar rooted binary trees with the internal nodes colored by the elements of S, graded by the number of leaves of a tree. We show that it has a unique structure of coassociative coalgebra such that the coproduct satisfies the unital infinitesimal condition with each magmatic product, and prove an analog of Aguiar-Sottile formula in this context, describing the coproduct in terms of the Moebius basis for the Tamari order. The last result allows us to compute the subspace of primitive elements of any unital infinitesimal S-magmatic bialgebra. As an example, we construct a set of generators of the dual of Pilaud and Pons bialgebra of integer relations and compute an explicit basis of its subspace of primitive elements. | math | 0 |
Title: Nonlinear Young integrals via fractional calculus
Abstract: For H\"older continuous functions $W(t,x)$ and $\varphi_t$, we define nonlinear integral $\int_a^b W(dt, \varphi_t)$ via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients. We also define iterated nonlinear integrals. | math | 1 |
Title: An exact test for renewal increasing mean residual life
Abstract: In this paper, we develop an exact test for testing exponentiality against renewal increasing mean residual life class. Pitman's asymptotic efficacy value shows that our test perform well. Some numerical results are presented to demonstrate the performance of the testing method. We also discuss how the proposed method incorporates the right censored observations. | math | 1 |
Title: Fundamental Theorem of Projective Geometry over Semirings
Abstract: We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry | math | 1 |
Title: Close to Human-Level Agreement: Tracing Journeys of Violent Speech in Incel Posts with GPT-4-Enhanced Annotations
Abstract: This study investigates the prevalence of violent language on incels.is. It evaluates GPT models (GPT-3.5 and GPT-4) for content analysis in social sciences, focusing on the impact of varying prompts and batch sizes on coding quality for the detection of violent speech. We scraped over 6.9M posts from incels.is and categorized a random sample into non-violent, explicitly violent, and implicitly violent content. Two human coders annotated 3,028 posts, which we used to tune and evaluate GPT-3.5 and GPT-4 models across different prompts and batch sizes regarding coding reliability. The best-performing GPT-4 model annotated an additional 30,000 posts for further analysis. Our findings indicate an overall increase in violent speech overtime on incels.is, both at the community and individual level, particularly among more engaged users. While directed violent language decreases, non-directed violent language increases, and self-harm content shows a decline, especially after 2.5 years of user activity. We find substantial agreement between both human coders (K = .65), while the best GPT-4 model yields good agreement with both human coders (K = 0.54 for Human A and K = 0.62 for Human B). Weighted and macro F1 scores further support this alignment. Overall, this research provides practical means for accurately identifying violent language at a large scale that can aid content moderation and facilitate next-step research into the causal mechanism and potential mitigations of violent expression and radicalization in communities like incels.is. | cs | 0 |
Title: Multi-Source Domain Adaptation with Transformer-based Feature Generation for Subject-Independent EEG-based Emotion Recognition
Abstract: Although deep learning-based algorithms have demonstrated excellent performance in automated emotion recognition via electroencephalogram (EEG) signals, variations across brain signal patterns of individuals can diminish the model's effectiveness when applied across different subjects. While transfer learning techniques have exhibited promising outcomes, they still encounter challenges related to inadequate feature representations and may overlook the fact that source subjects themselves can possess distinct characteristics. In this work, we propose a multi-source domain adaptation approach with a transformer-based feature generator (MSDA-TF) designed to leverage information from multiple sources. The proposed feature generator retains convolutional layers to capture shallow spatial, temporal, and spectral EEG data representations, while self-attention mechanisms extract global dependencies within these features. During the adaptation process, we group the source subjects based on correlation values and aim to align the moments of the target subject with each source as well as within the sources. MSDA-TF is validated on the SEED dataset and is shown to yield promising results. | cs | 0 |
Title: Bayesian Intrinsic Groupwise Image Registration: Unsupervised Disentanglement of Anatomy and Geometry
Abstract: This article presents a general Bayesian learning framework for multi-modal groupwise registration on medical images. The method builds on probabilistic modelling of the image generative process, where the underlying common anatomy and geometric variations of the observed images are explicitly disentangled as latent variables. Thus, groupwise registration is achieved through the solution to Bayesian inference. We propose a novel hierarchical variational auto-encoding architecture to realize the inference procedure of the latent variables, where the registration parameters can be calculated in a mathematically interpretable fashion. Remarkably, this new paradigm can learn groupwise registration in an unsupervised closed-loop self-reconstruction process, sparing the burden of designing complex intensity-based similarity measures. The computationally efficient disentangled architecture is also inherently scalable and flexible, allowing for groupwise registration on large-scale image groups with variable sizes. Furthermore, the inferred structural representations from disentanglement learning are capable of capturing the latent anatomy of the observations with visual semantics. Extensive experiments were conducted to validate the proposed framework, including four datasets from cardiac, brain and abdominal medical images. The results have demonstrated the superiority of our method over conventional similarity-based approaches in terms of accuracy, efficiency, scalability and interpretability. | cs | 0 |
Title: Coded Beam Training
Abstract: In extremely large-scale multiple input multiple output (XL-MIMO) systems for future sixth-generation (6G) communications, codebook-based beam training stands out as a promising technology to acquire channel state information (CSI). Despite their effectiveness, when the pilot overhead is limited, existing beam training methods suffer from significant achievable rate degradation for remote users with low signal-to-noise ratio (SNR). To tackle this challenge, leverging the error-correcting capability of channel codes, we introduce channel coding theory into hierarchical beam training to extend the coverage area. Specifically, we establish the duality between hierarchical beam training and channel coding, and the proposed coded beam training scheme serves as a general framework. Then, we present two specific implementations exemplified by coded beam training methods based on Hamming codes and convolutional codes, during which the beam encoding and decoding processes are refined respectively to better accommodate to the beam training problem. Simulation results have demonstrated that, the proposed coded beam training method can enable reliable beam training performance for remote users with low SNR, while keeping training overhead low. | cs | 0 |
Title: List-Coloring Packing and Correspondence-Coloring Packing of Planar Graphs
Abstract: For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $\varphi_1,\cdots,\varphi_k$ such that $\varphi_i(v)\ne\varphi_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let $\chi^{\star}_{\ell}(G)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $|L(v)|=k$ for all $v$. Let $\mathcal{P}_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) $\chi^{\star}_{\ell}(G)\le 8$ for all $G\in \mathcal{P}_3$ and (ii) $\chi^{\star}_{\ell}(G)\le 5$ for all $G\in \mathcal{P}_4$ and (iii) $\chi^{\star}_{\ell}(G)\le 4$ for all $G\in \mathcal{P}_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also construct outerplanar graphs $G$ such that $\chi^{\star}_{\ell}(G)=4$, which matches the known upper bound $\chi^{\star}_{\ell}(G)\le 4$ for all outerplanar graphs. Finally, we consider the analogue of $\chi^{\star}_{\ell}$ for correspondence coloring, $\chi^{\star}_c$. In fact, all bounds stated above for $\chi^{\star}_{\ell}$ also hold for $\chi^{\star}_c$. | math | 0 |
Title: A note on a question of Shioda about integral sections
Abstract: We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some equations. | math | 0 |
Title: Training Single-Layer Morphological Perceptron Using Convex-Concave Programming
Abstract: This paper concerns the training of a single-layer morphological perceptron using disciplined convex-concave programming (DCCP). We introduce an algorithm referred to as K-DDCCP, which combines the existing single-layer morphological perceptron (SLMP) model proposed by Ritter and Urcid with the weighted disciplined convex-concave programming (WDCCP) algorithm by Charisopoulos and Maragos. The proposed training algorithm leverages the disciplined convex-concave procedure (DCCP) and formulates a non-convex optimization problem for binary classification. To tackle this problem, the constraints are expressed as differences of convex functions, enabling the application of the DCCP package. The experimental results confirm the effectiveness of the K-DDCCP algorithm in solving binary classification problems. Overall, this work contributes to the field of morphological neural networks by proposing an algorithm that extends the capabilities of the SLMP model. | cs | 0 |
Title: On semigroups of orientation-preserving partial permutations with restricted range
Abstract: Let $\Omega_n$ be a finite chain with $n$ elements $(n\in\mathbb{N})$, and let $\mathcal{POPI}_{n}$ be the semigroup of all injective orientation-preserving partial transformations of $\Omega_n$. In this paper, for any nonempty subset $Y$ of $\Omega_n$, we consider the subsemigroup $\mathcal{POPI}_{n}(Y)$ of $\mathcal{POPI}_{n}$ of all transformations with range contained in $Y$. We describe the Green's relations and study the regularity of $\mathcal{POPI}_{n}(Y)$. Moreover, we calculate the rank of $\mathcal{POPI}_{n}(Y)$ and determine when two semigroups of this type are isomorphic. | math | 0 |
Title: Hamiltonicity of Schrijver graphs and stable Kneser graphs
Abstract: For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers $k\geq 1$, $s\geq 2$, and $n \geq sk+1$, the $s$-stable Kneser graph $S(n,k,s)$ has as vertices all $k$-element subsets of $[n]$ in which any two elements are in cyclical distance at least $s$. We prove that all the graphs $S(n,k,s)$, in particular Schrijver graphs $S(n,k)=S(n,k,2)$, admit a Hamilton cycle that can be computed in time $\mathcal{O}(n)$ per generated vertex. | math | 0 |
Title: Non-trivial higher homotopy of first-order theories
Abstract: Let $T$ be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf{Mod}(T)$ to be homotopically equivalent to $\mathbb{CP}^\infty$. In particular, $\pi_2(\lvert\mathsf{Mod}(T)\rvert)=\mathbb{Z}$, which answers a question in our previous work. The computation is based on Connes' cycle category $\Lambda$. | math | 0 |
Title: On some permanence properties of (derived) splinters
Abstract: We show that Noetherian splinters ascend under essentially \'etale homomorphisms. Along the way, we also prove that the henselization of a Noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (non-excellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. Our results provide evidence for a strengthening of the direct summand theorem, namely that regular maps preserve the splinter property. | math | 1 |
Title: Multiplicative and additive compounds via Kronecker products and Kronecker sums
Abstract: Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use. Here, we derive new formulas for the multiplicative and additive compounds of a matrix using Kronecker products and sums. This provides a new approach to matrix compounds based on the well-known and powerful theory of Kronecker products and sums. We demonstrate several applications of these new formulas, including deriving a new expression for the additive compound of the product of two matrices. | math | 0 |
Title: InstructTA: Instruction-Tuned Targeted Attack for Large Vision-Language Models
Abstract: Large vision-language models (LVLMs) have demonstrated their incredible capability in image understanding and response generation. However, this rich visual interaction also makes LVLMs vulnerable to adversarial examples. In this paper, we formulate a novel and practical gray-box attack scenario that the adversary can only access the visual encoder of the victim LVLM, without the knowledge of its prompts (which are often proprietary for service providers and not publicly available) and its underlying large language model (LLM). This practical setting poses challenges to the cross-prompt and cross-model transferability of targeted adversarial attack, which aims to confuse the LVLM to output a response that is semantically similar to the attacker's chosen target text. To this end, we propose an instruction-tuned targeted attack (dubbed InstructTA) to deliver the targeted adversarial attack on LVLMs with high transferability. Initially, we utilize a public text-to-image generative model to "reverse" the target response into a target image, and employ GPT-4 to infer a reasonable instruction $\boldsymbol{p}^\prime$ from the target response. We then form a local surrogate model (sharing the same visual encoder with the victim LVLM) to extract instruction-aware features of an adversarial image example and the target image, and minimize the distance between these two features to optimize the adversarial example. To further improve the transferability, we augment the instruction $\boldsymbol{p}^\prime$ with instructions paraphrased from an LLM. Extensive experiments demonstrate the superiority of our proposed method in targeted attack performance and transferability. | cs | 0 |
Title: Approximating the Shapley Value without Marginal Contributions
Abstract: The Shapley value, which is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, has recently been used intensively in explainable artificial intelligence. Its meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley value, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contribution. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods. | cs | 0 |
Title: Continuity of composition operators in Sobolev spaces
Abstract: We prove that all the composition operators $T_f(g):= f\circ g$, which take the Adams-Frazier space $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, are continuous mappings from $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, for every integer $m\geq 2$ and every real number $1\leq p<+\infty$. The same automatic continuity property holds for Sobolev spaces $W^m_p(\mathbb{R}^n)$ for $m\geq 2$ and $1\leq p<+\infty$. | math | 1 |
Title: Formal multiple Eisenstein series and their derivations
Abstract: We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an $\mathfrak{sl}_2$-algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms. | math | 0 |
Title: Logit-Q Dynamics for Efficient Learning in Stochastic Teams
Abstract: We present two logit-Q learning dynamics combining the classical and independent log-linear learning updates with an on-policy value iteration update for efficient learning in stochastic games. We show that the logit-Q dynamics presented reach (near) efficient equilibrium in stochastic teams. We quantify a bound on the approximation error. We also show the rationality of the logit-Q dynamics against agents following pure stationary strategies and the convergence of the dynamics in stochastic games where the reward functions induce potential games, yet only a single agent controls the state transitions beyond stochastic teams. The key idea is to approximate the dynamics with a fictional scenario where the Q-function estimates are stationary over finite-length epochs only for analysis. We then couple the dynamics in the main and fictional scenarios to show that these two scenarios become more and more similar across epochs due to the vanishing step size. | cs | 0 |
Title: Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials
Abstract: We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-\Delta)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough. | math | 0 |
Title: Complex structures on stratified Lie algebras
Abstract: This paper investigates some properties of complex structures on Lie algebras. In particular, we focus on $\textit{nilpotent}$ $\textit{complex structures}$ that are characterized by a suitable $J$-invariant ascending or descending central series $\mathfrak{d}^j$ and $\mathfrak{d}_j$ respectively. In this article, we introduce a new descending series $\mathfrak{p}_j$ and use it to give proof of a new characterization of nilpotent complex structures. We examine also whether nilpotent complex structures on stratified Lie algebras preserve the strata. We find that there exists a $J$-invariant stratification on a step $2$ nilpotent Lie algebra with a complex structure. | math | 1 |
Title: Shrinking unit: a Graph Convolution-Based Unit for CNN-like 3D Point Cloud Feature Extractors
Abstract: 3D point clouds have attracted increasing attention in architecture, engineering, and construction due to their high-quality object representation and efficient acquisition methods. Consequently, many point cloud feature detection methods have been proposed in the literature to automate some workflows, such as their classification or part segmentation. Nevertheless, the performance of point cloud automated systems significantly lags behind their image counterparts. While part of this failure stems from the irregularity, unstructuredness, and disorder of point clouds, which makes the task of point cloud feature detection significantly more challenging than the image one, we argue that a lack of inspiration from the image domain might be the primary cause of such a gap. Indeed, given the overwhelming success of Convolutional Neural Networks (CNNs) in image feature detection, it seems reasonable to design their point cloud counterparts, but none of the proposed approaches closely resembles them. Specifically, even though many approaches generalise the convolution operation in point clouds, they fail to emulate the CNNs multiple-feature detection and pooling operations. For this reason, we propose a graph convolution-based unit, dubbed Shrinking unit, that can be stacked vertically and horizontally for the design of CNN-like 3D point cloud feature extractors. Given that self, local and global correlations between points in a point cloud convey crucial spatial geometric information, we also leverage them during the feature extraction process. We evaluate our proposal by designing a feature extractor model for the ModelNet-10 benchmark dataset and achieve 90.64% classification accuracy, demonstrating that our innovative idea is effective. Our code is available at github.com/albertotamajo/Shrinking-unit. | cs | 1 |
Title: Relating the Roe algebra of a space to the uniform Roe algebras of its discretizations
Abstract: The Roe algebra $C^*(X)$ is a non-commutative $C^*$-algebra reflecting metric properties of a space $X$, and it is interesting to understand relation between the Roe algebra of $X$ and the uniform Roe algebra of its discretizations. Here we construct, for a simplicial space $X$, a continuous field of $C^*$-algebras over $\mathbb N\cup\{\infty\}$ with the fibers over finite points the uniform $C^*$-algebras of discretizations of $X$, and the fiber over $\infty$ the Roe algebra of $X$. We also construct the direct limit of the uniform Roe algebras of discretizations and its embedding into the Roe algebra of $X$. | math | 0 |
Title: A small time approximation for the solution to the Zakai Equation
Abstract: We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov's type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds. | math | 1 |
Title: Convergence of Stochastic Gradient Descent for PCA
Abstract: We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i.i.d. data points in $\reals^d$. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in \cite{hardt2014noisy}. Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap. | cs | 1 |
Title: The bar derived category of a curved dg algebra
Abstract: Curved A-infinity algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A-infinity algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra. | math | 1 |
Title: Free resolution of the logarithmic derivation modules of close to free arrangements
Abstract: This paper studies the algebraic structure of a new class of hyperplane arrangement $A$ obtained by deleting two hyperplanes from a free arrangement. We provide information on the minimal free resolutions of the logarithmic derivation module of $A$, which can be used to compute a lower bound for the graded Betti numbers of the resolution. Specifically, for the three-dimensional case, we determine the minimal free resolution of the logarithmic derivation module of $A$. We present illustrative examples of our main theorems to provide insights into the relationship between algebraic and combinatorial properties for close-to-free arrangements. | math | 0 |
Title: Self-adjointness of the Gaffney Laplacian on vector bundles
Abstract: We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator. | math | 1 |
Title: Entropy and sampling numbers of classes of ridge functions
Abstract: We study properties of ridge functions $f(x)=g(a\cdot x)$ in high dimensions $d$ from the viewpoint of approximation theory. The considered function classes consist of ridge functions such that the profile $g$ is a member of a univariate Lipschitz class with smoothness $\alpha > 0$ (including infinite smoothness), and the ridge direction $a$ has $p$-norm $\|a\|_p \leq 1$. First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in $L_{\infty}$. We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and face two extreme cases. In case $p=2$, sampling ridge functions on the Euclidean unit ball faces the curse of dimensionality. It is thus as difficult as sampling general multivariate Lipschitz functions, a result in sharp contrast to the result on entropy numbers. When we additionally assume that all feasible profiles have a first derivative uniformly bounded away from zero in the origin, then the complexity of sampling ridge functions reduces drastically to the complexity of sampling univariate Lipschitz functions. In between, the sampling problem's degree of difficulty varies, depending on the values of $\alpha$ and $p$. Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Wo\'zniakowski. | math | 1 |
Title: Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs
Abstract: The enumeration of Hamiltonian cycles on 2n*2n grids of nodes is a longstanding problem in combinatorics. Previous work has concentrated on counting all cycles. The current work enumerates nonisomorphic cycles -- that is, the number of isomorphism classes (up to all symmetry operations of the square). It is shown that the matrix method used previously can be modified to count cycles with all combinations of reflective and 180-degree rotational symmetry. Cycles with 90-degree rotational symmetry were counted by a direct search, using a modification of Knuth's Dancing Links algorithm. From these counts, the numbers of nonisomorphic cycles were calculated for n<=10. | math | 1 |
Title: Diabetic Retinopathy Using Gaussian Filter
Abstract: The retina is an essential component of the visual system, and maintaining eyesight depends on the timely and correct detection of disorders. This research specifically addresses the early-stage detection and severity classification of diabetic retinopathy (DR), a serious public health hazard. We compare the results of different deep learning models such as InceptionV3, DenseNet121 and other CNN based models by using different image filters, such as Gaussian, grayscale and Gabor. These models could detect subtle pathological alterations and use that information to estimate the risk of retinal illnesses. The objective is to improve the diagnostic processes for diabetic retinopathy, the primary cause of diabetes-related blindness, by utilizing deep learning models. A comparative analysis between Greyscale, Gaussian and Gabor filters has been provided after applying these filters on the retinal images. The Gaussian filter resulted to be the most promising filter giving the best accuracies for all the models. The best performing model was InceptionV3 which gave an accuracy of 96% on Gaussian images, therefore Gaussian filter emerged as our most promising filter. | cs | 0 |
Title: What You See is What You GAN: Rendering Every Pixel for High-Fidelity Geometry in 3D GANs
Abstract: 3D-aware Generative Adversarial Networks (GANs) have shown remarkable progress in learning to generate multi-view-consistent images and 3D geometries of scenes from collections of 2D images via neural volume rendering. Yet, the significant memory and computational costs of dense sampling in volume rendering have forced 3D GANs to adopt patch-based training or employ low-resolution rendering with post-processing 2D super resolution, which sacrifices multiview consistency and the quality of resolved geometry. Consequently, 3D GANs have not yet been able to fully resolve the rich 3D geometry present in 2D images. In this work, we propose techniques to scale neural volume rendering to the much higher resolution of native 2D images, thereby resolving fine-grained 3D geometry with unprecedented detail. Our approach employs learning-based samplers for accelerating neural rendering for 3D GAN training using up to 5 times fewer depth samples. This enables us to explicitly "render every pixel" of the full-resolution image during training and inference without post-processing superresolution in 2D. Together with our strategy to learn high-quality surface geometry, our method synthesizes high-resolution 3D geometry and strictly view-consistent images while maintaining image quality on par with baselines relying on post-processing super resolution. We demonstrate state-of-the-art 3D gemetric quality on FFHQ and AFHQ, setting a new standard for unsupervised learning of 3D shapes in 3D GANs. | cs | 0 |
Title: Constrained quantization for the Cantor distribution with a family of constraints
Abstract: In this paper, for a given family of constraints and the classical Cantor distribution we determine the optimal sets of $n$-points, $n$th constrained quantization errors for all positive integers $n$. We also calculate the constrained quantization dimension and the constrained quantization coefficient, and see that the constrained quantization dimension $D(P)$ exists as a finite positive number, but the $D(P)$-dimensional constrained quantization coefficient does not exist. | math | 0 |
Title: Post-Processed Posteriors for Banded Covariances
Abstract: We consider Bayesian inference of banded covariance matrices and propose a post-processed posterior. The post-processing of the posterior consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior which does not satisfy any structural restrictions. In the second step, the posterior samples are transformed to satisfy the structural restriction through a post-processing function. The conceptually straightforward procedure of the post-processed posterior makes its computation efficient and can render interval estimators of functionals of covariance matrices. We show that it has nearly optimal minimax rates for banded covariances among all possible pairs of priors and post-processing functions. Furthermore, we prove that the expected coverage probability of the $(1-\alpha)100\%$ highest posterior density region of the post-processed posterior is asymptotically $1-\alpha$ with respect to a conventional posterior distribution. It implies that the highest posterior density region of the post-processed posterior is, on average, a credible set of a conventional posterior. The advantages of the post-processed posterior are demonstrated by a simulation study and a real data analysis. | math | 1 |
Title: Shadow Generation with Decomposed Mask Prediction and Attentive Shadow Filling
Abstract: Image composition refers to inserting a foreground object into a background image to obtain a composite image. In this work, we focus on generating plausible shadows for the inserted foreground object to make the composite image more realistic. To supplement the existing small-scale dataset, we create a large-scale dataset called RdSOBA with rendering techniques. Moreover, we design a two-stage network named DMASNet with decomposed mask prediction and attentive shadow filling. Specifically, in the first stage, we decompose shadow mask prediction into box prediction and shape prediction. In the second stage, we attend to reference background shadow pixels to fill the foreground shadow. Abundant experiments prove that our DMASNet achieves better visual effects and generalizes well to real composite images. | cs | 0 |
Title: Principle of Relevant Information for Graph Sparsification
Abstract: Graph sparsification aims to reduce the number of edges of a graph while maintaining its structural properties. In this paper, we propose the first general and effective information-theoretic formulation of graph sparsification, by taking inspiration from the Principle of Relevant Information (PRI). To this end, we extend the PRI from a standard scalar random variable setting to structured data (i.e., graphs). Our Graph-PRI objective is achieved by operating on the graph Laplacian, made possible by expressing the graph Laplacian of a subgraph in terms of a sparse edge selection vector $\mathbf{w}$. We provide both theoretical and empirical justifications on the validity of our Graph-PRI approach. We also analyze its analytical solutions in a few special cases. We finally present three representative real-world applications, namely graph sparsification, graph regularized multi-task learning, and medical imaging-derived brain network classification, to demonstrate the effectiveness, the versatility and the enhanced interpretability of our approach over prevalent sparsification techniques. Code of Graph-PRI is available at https://github.com/SJYuCNEL/PRI-Graphs | cs | 1 |
Title: The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry
Abstract: In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric K\"ahler manifolds as their torifications.We introduce a dually flat structure and the associated Bregman divergence on the boundary from the viewpoint of toric K\"ahler geometry. We show a continuity and a generalized Pythagorean theorem for the divergence on the boundary. We also provide a characterization for a toric K\"ahler manifold to become a torification of a mixture family on a finite set. | math | 0 |
Title: Embedding Hypertrees into Steiner Triple Systems
Abstract: In this paper we are interested in the following question: Given an arbitrary Steiner triple system $S$ on $m$ vertices and any 3-uniform hypertree $T$ on $n$ vertices, is it necessary that $S$ contains $T$ as a subgraph provided $m \geq (1+\mu)n$? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive. | math | 1 |
Title: Predicting Post-Route Quality of Results Estimates for HLS Designs using Machine Learning
Abstract: Machine learning (ML) has been widely used to improve the predictability of EDA tools. The use of CAD tools that express designs at higher levels of abstraction makes machine learning even more important to highlight the performance of various design steps. Behavioral descriptions used during the high-level synthesis (HLS) are completely technology independent making it hard for designers to interpret how changes in the synthesis options affect the resultant circuit. FPGA design flows are completely embracing HLS based methodologies so that software engineers with almost no hardware design skills can easily use their tools. HLS tools allow design space exploration by modifying synthesis options, however, they lack accuracy in the Quality of Results (QoR) reported right after HLS. This lack of correctness results in sub-optimal designs with problems in timing closure. This paper presents a robust ML based design flow that can accurately predict post-route QoR for a given behavioral description without the need to synthesize the design. The model is an important design exploration tool where a designer can quickly view the impact on overall design quality when local and global optimization directives are changed. The proposed methodology presents two strong advantages: (i) Accurate prediction of the design quality (QoR), and (ii) complete elimination of the need to execute high-level synthesis for each design option. We predict three post route parameters, (i). Area, (ii). Latency and (iii). Clock Period of a design just by analyzing the high level behavioral code and some intermediate representation codes. We have integrated the methodology with Xilinx HLS tools and have demonstrated accurate estimation on a variety of FPGA families. Our estimated results are within 10\% of actual computed values | cs | 1 |
Title: Clairaut Anti-invariant Riemannian maps from/to Kähler manifolds admitting Ricci soliton
Abstract: The aim of this article is to study the Clairaut anti-invariant Riemannian maps from/to K\"ahler manifolds admitting Ricci solitons. We find the curvature relations and calculate the Ricci tensor under different conditions. We obtain conditions for the range and kernel of these maps to be Einstein. Next, we find the scalar curvature for range. Finally, we give some non-trivial examples of Clairaut anti-invariant Riemannian maps from/to K\"ahler manifolds admitting Ricci solitons. | math | 0 |
Title: Explicit construction of a plane sextic model for genus-five Howe curves, II
Abstract: A Howe curve is defined as the normalization of the fiber product over a projective line of two hyperelliptic curves. Howe curves are very useful to produce important classes of curves over fields of positive characteristic, e.g., maximal, superspecial, or supersingular ones. Determining their feasible equations explicitly is a basic problem, and it has been solved in the hyperelliptic case and in the non-hyperelliptic case with genus not greater than $4$. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus $5$. We also determine the number and type of singularities on our sextic model, and prove that the singularities are generically $4$ double points. Our results together with Moriya-Kudo's recent ones imply that for each $s \in \{2,3,4,5\}$, there exists a non-hyperellptic curve $H$ of genus $5$ with $\mathrm{Aut}(H) \supset \mathbf{V}_4$ such that its associated plane sextic has $s$ double points. | math | 0 |
Title: Computing higher symplectic capacities I
Abstract: We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated filtered L-infinity algebras and prove that the resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As sample applications, we produce new previously inaccessible obstructions for stabilized embeddings of ellipsoids and polydisks, and we give new counts of curves with tangency constraints. | math | 1 |
Title: Assessing the Impact of a User-Item Collaborative Attack on Class of Users
Abstract: Collaborative Filtering (CF) models lie at the core of most recommendation systems due to their state-of-the-art accuracy. They are commonly adopted in e-commerce and online services for their impact on sales volume and/or diversity, and their impact on companies' outcome. However, CF models are only as good as the interaction data they work with. As these models rely on outside sources of information, counterfeit data such as user ratings or reviews can be injected by attackers to manipulate the underlying data and alter the impact of resulting recommendations, thus implementing a so-called shilling attack. While previous works have focused on evaluating shilling attack strategies from a global perspective paying particular attention to the effect of the size of attacks and attacker's knowledge, in this work we explore the effectiveness of shilling attacks under novel aspects. First, we investigate the effect of attack strategies crafted on a target user in order to push the recommendation of a low-ranking item to a higher position, referred to as user-item attack. Second, we evaluate the effectiveness of attacks in altering the impact of different CF models by contemplating the class of the target user, from the perspective of the richness of her profile (i.e., cold v.s. warm user). Finally, similar to previous work we contemplate the size of attack (i.e., the amount of fake profiles injected) in examining their success. The results of experiments on two widely used datasets in business and movie domains, namely Yelp and MovieLens, suggest that warm and cold users exhibit contrasting behaviors in datasets with different characteristics. | cs | 1 |
Title: Global Invariant Branches of Non-degenerate Foliations on Projective Toric Surfaces
Abstract: We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type foliations in terms of "non-degeneracy" conditions, associated to Newton polygons. We also give a description of the relationship between invariant algebraic curves and isolated invariant branches, valid for the case of toric type, by means of the following dichotomy. Either there is a rational first integral and there are no isolated invariant branches or we have only finitely many global invariant curves, all of them extending isolated invariant branches. | math | 1 |
Title: SENS3: Multisensory Database of Finger-Surface Interactions and Corresponding Sensations
Abstract: The growing demand for natural interactions with technology underscores the importance of achieving realistic touch sensations in digital environments. Realizing this goal highly depends on comprehensive databases of finger-surface interactions, which need further development. Here, we present SENS3, an extensive open-access repository of multisensory data acquired from fifty surfaces when two participants explored them with their fingertips through static contact, pressing, tapping, and sliding. SENS3 encompasses high-fidelity visual, audio, and haptic information recorded during these interactions, including videos, sounds, contact forces, torques, positions, accelerations, skin temperature, heat flux, and surface photographs. Additionally, it incorporates thirteen participants' psychophysical sensation ratings while exploring these surfaces freely. We anticipate that SENS3 will be valuable for advancing multisensory texture rendering, user experience development, and touch sensing in robotics. | cs | 0 |
Title: Summing a family of generalized Pell numbers
Abstract: A new family of generalized Pell numbers was recently introduced and studied by Br\'od \cite{Dorota}. These number possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power $P_n^l$ is expressed as a linear combination of $P_{mn}$. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter $R=2^r$, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times. | math | 1 |
Title: Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone
Abstract: We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example. | math | 0 |
Title: Inferring relevant features: from QFT to PCA
Abstract: In many-body physics, renormalization techniques are used to extract aspects of a statistical or quantum state that are relevant at large scale, or for low energy experiments. Recent works have proposed that these features can be formally identified as those perturbations of the states whose distinguishability most resist coarse-graining. Here, we examine whether this same strategy can be used to identify important features of an unlabeled dataset. This approach indeed results in a technique very similar to kernel PCA (principal component analysis), but with a kernel function that is automatically adapted to the data, or "learned". We test this approach on handwritten digits, and find that the most relevant features are significantly better for classification than those obtained from a simple gaussian kernel. | cs | 1 |
Title: Ordered Ramsey numbers of powers of paths
Abstract: Given two vertex-ordered graphs $G$ and $H$, the ordered Ramsey number $R_<(G,H)$ is the smallest $N$ such that whenever the edges of a vertex-ordered complete graph $K_N$ are red/blue-coloured, then there is a red (ordered) copy of $G$ or a blue (ordered) copy of $H$. Let $P_n^t$ denote the $t$-th power of a monotone path on $n$ vertices. The ordered Ramsey numbers of powers of paths have been extensively studied. We prove that there exists an absolute constant $C$ such that $R_<(K_s,P_n^t)\leq R(K_s,K_t)^{C} \cdot n$ holds for all $s,t,n$, which is tight up to the value of $C$. As a corollary, we obtain that there is an absolute constant $C$ such that $R_<(K_n,P_n^t)\leq n^{Ct}$. These results resolve a problem and a conjecture of Gishboliner, Jin and Sudakov. Furthermore, we show that $R_<(P_n^t,P_n^t)\leq n^{4+o(1)}$ for any fixed $t$. This answers questions of Balko, Cibulka, Kr\'al and Kyn\v{c}l, and of Gishboliner, Jin and Sudakov. | math | 0 |
Title: Recursive sequences attached to modular representations of finite groups
Abstract: The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic, in the sense that the non-projective summands of $M^{\otimes n}$ fall into finitely many orbits under the action of the syzygy operator $\Omega$. Similarly, we prove that these dimension sequences are eventually linearly recursive when $M$ is what we term $\Omega^{+}$-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences. | math | 1 |
Title: Dirac geometry II: Coherent cohomology
Abstract: Whatever it is that animates anima and breathes life into higher algebra, this something leaves its trace in the structure of a Dirac ring on the homotopy groups of a commutative algebra in spectra. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we first embed this category in the larger infinity-category of Dirac stacks, which also contains formal Dirac schemes. We next develop the coherent cohomology of Dirac stacks, which amounts to a functor that to a Dirac stack X assigns a presentably symmetric monoidal stable infinity-category QCoh(X) of quasi-coherent sheaves together with a compatible t-structure. Finally, as applications of the general theory to stable homotopy theory, we use Quillen's theorem on complex cobordism and Milnor's theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to MU and F_p in terms of their functors of points. In the appendix, we develop a rudimentary theory of accessible presheaves of anima on coaccessible infinity-categories. | math | 0 |
Title: Variance Reduction is an Antidote to Byzantines: Better Rates, Weaker Assumptions and Communication Compression as a Cherry on the Top
Abstract: Byzantine-robustness has been gaining a lot of attention due to the growth of the interest in collaborative and federated learning. However, many fruitful directions, such as the usage of variance reduction for achieving robustness and communication compression for reducing communication costs, remain weakly explored in the field. This work addresses this gap and proposes Byz-VR-MARINA - a new Byzantine-tolerant method with variance reduction and compression. A key message of our paper is that variance reduction is key to fighting Byzantine workers more effectively. At the same time, communication compression is a bonus that makes the process more communication efficient. We derive theoretical convergence guarantees for Byz-VR-MARINA outperforming previous state-of-the-art for general non-convex and Polyak-Lojasiewicz loss functions. Unlike the concurrent Byzantine-robust methods with variance reduction and/or compression, our complexity results are tight and do not rely on restrictive assumptions such as boundedness of the gradients or limited compression. Moreover, we provide the first analysis of a Byzantine-tolerant method supporting non-uniform sampling of stochastic gradients. Numerical experiments corroborate our theoretical findings. | cs | 1 |
Title: The Chern class for $K_3$ and the cyclic quantum dilogarithm
Abstract: In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in their recent paper in Ann. Sci. \'{E}cole Normale Sup. that $R_\zeta=c_\zeta^2$, where $R_\zeta$ is their map on $K_3$ defined using the cyclic quantum dilogarithm and $c_\zeta$ is the Chern class map on $K_3$. | math | 1 |
Title: On the parametrized Tate construction and two theories of real $p$-cyclotomic spectra
Abstract: We give a new formula for $p$-typical real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for $p$-typical topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this, we give a new definition of the $\infty$-category of real $p$-cyclotomic spectra that replaces the usage of genuinely equivariant dihedral spectra with the parametrized Tate construction $(-)^{t_{C_2} \mu_p}$ associated to the dihedral group $D_{2p} = \mu_p \rtimes C_2$. We then define a $p$-typical and $\infty$-categorical version of H{\o}genhaven's $O(2)$-orthogonal cyclotomic spectra, construct a forgetful functor relating the two theories, and show that this functor restricts to an equivalence between full subcategories of appropriately bounded below objects. | math | 1 |
Title: Level-set based design of Wang tiles for modelling complex microstructures
Abstract: Microstructural geometry plays a critical role in the response of heterogeneous materials. Consequently, methods for generating microstructural samples are increasingly crucial to advanced numerical analyses. We extend Sonon et al.'s unified framework, developed originally for generating particulate and foam-like microstructural geometries of Periodic Unit Cells, to non-periodic microstructural representations based on the formalism of Wang tiles. This formalism has been recently proposed in order to generalize the Periodic Unit Cell approach, enabling a fast synthesis of arbitrarily large, stochastic microstructural samples from a handful of domains with predefined compatibility constraints. However, a robust procedure capable of designing complex, three-dimensional, foam-like and cellular morphologies of Wang tiles has not yet been proposed. This contribution fills the gap by significantly broadening the applicability of the tiling concept. Since the original Sonon et al.'s framework builds on a random sequential addition of particles enhanced with an implicit representation of particle boundaries by the level-set field, we first devise an analysis based on a connectivity graph of a tile set, resolving the question where a particle should be copied when it intersects a tile boundary. Next, we introduce several modifications to the original algorithm that are necessary to ensure microstructural compatibility in the generalized periodicity setting of Wang tiles. Having established a universal procedure for generating tile morphologies, we compare strictly aperiodic and stochastic sets with the same cardinality in terms of reducing the artificial periodicity in reconstructed microstructural samples. We demonstrate the superiority of the vertex-defined tile sets for two-dimensional problems and illustrate the capabilities of the algorithm with two- and three-dimensional examples. | cs | 1 |
Title: A Tutorial on Extremely Large-Scale MIMO for 6G: Fundamentals, Signal Processing, and Applications
Abstract: Extremely large-scale multiple-input-multiple-output (XL-MIMO), which offers vast spatial degrees of freedom, has emerged as a potentially pivotal enabling technology for the sixth generation (6G) of wireless mobile networks. With its growing significance, both opportunities and challenges are concurrently manifesting. This paper presents a comprehensive survey of research on XL-MIMO wireless systems. In particular, we introduce four XL-MIMO hardware architectures: uniform linear array (ULA)-based XL-MIMO, uniform planar array (UPA)-based XL-MIMO utilizing either patch antennas or point antennas, and continuous aperture (CAP)-based XL-MIMO. We comprehensively analyze and discuss their characteristics and interrelationships. Following this, we introduce several electromagnetic characteristics and general distance boundaries in XL-MIMO. Given the distinct electromagnetic properties of near-field communications, we present a range of channel models to demonstrate the benefits of XL-MIMO. We further discuss and summarize signal processing schemes for XL-MIMO. It is worth noting that the low-complexity signal processing schemes and deep learning empowered signal processing schemes are reviewed and highlighted to promote the practical implementation of XL-MIMO. Furthermore, we explore the interplay between XL-MIMO and other emergent 6G technologies. Finally, we outline several compelling research directions for future XL-MIMO wireless communication systems. | cs | 0 |
Title: Visual Relationship Forecasting in Videos
Abstract: Real-world scenarios often require the anticipation of object interactions in unknown future, which would assist the decision-making process of both humans and agents. To meet this challenge, we present a new task named Visual Relationship Forecasting (VRF) in videos to explore the prediction of visual relationships in a reasoning manner. Specifically, given a subject-object pair with H existing frames, VRF aims to predict their future interactions for the next T frames without visual evidence. To evaluate the VRF task, we introduce two video datasets named VRF-AG and VRF-VidOR, with a series of spatio-temporally localized visual relation annotations in a video. These two datasets densely annotate 13 and 35 visual relationships in 1923 and 13447 video clips, respectively. In addition, we present a novel Graph Convolutional Transformer (GCT) framework, which captures both object-level and frame-level dependencies by spatio-temporal Graph Convolution Network and Transformer. Experimental results on both VRF-AG and VRF-VidOR datasets demonstrate that GCT outperforms the state-of-the-art sequence modelling methods on visual relationship forecasting. | cs | 1 |
Title: Propagation of Input Tail Uncertainty in Rare-Event Estimation: A Light versus Heavy Tail Dichotomy
Abstract: We consider the estimation of small probabilities or other risk quantities associated with rare but catastrophic events. In the model-based literature, much of the focus has been devoted to efficient Monte Carlo computation or analytical approximation assuming the model is accurately specified. In this paper, we study a distinct direction on the propagation of model uncertainty and how it impacts the reliability of rare-event estimates. Specifically, we consider the basic setup of the exceedance of i.i.d. sum, and investigate how the lack of tail information of each input summand can affect the output probability. We argue that heavy-tailed problems are much more vulnerable to input uncertainty than light-tailed problems, reasoned through their large deviations behaviors and numerical evidence. We also investigate some approaches to quantify model errors in this problem using a combination of the bootstrap and extreme value theory, showing some positive outcomes but also uncovering some statistical challenges. | math | 0 |
Title: Modeling multiple taxis: tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence
Abstract: We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character. We prove global existence of weak solutions to a simplified model version and perform numerical simulations for the full setting under several phenotype switching and motility scenarios. We also compare (via simulations) this model with the corresponding haptotaxis-chemotaxis one featuring indirect chemorepellent production and provide a discussion about possible model extensions and mathematical challenges. | math | 1 |
Title: An extremal problem for the convolution of logarithmically concave functions
Abstract: A new position is introduced and studied for the convolution of log-concave functions, which may be regarded as a functional analogue of the maximum intersection position of convex bodies introduced and studied by Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our main result is a John-type theorem for the maximal intersection position of a pair of log-concave functions, including the corresponding decomposition of the identity. The main result holds under very weak assumptions on the functions; in particular, the functions considered may both have unbounded supports. | math | 0 |
Title: Constructing Approximate Single-Source Distance Sensitivity Oracles in Nearly Linear Time
Abstract: For an input graph $G=(V, E)$ and a source vertex $s \in V$, the \emph{$\alpha$-approximate vertex fault-tolerant distance sensitivity oracle} (\emph{$\alpha$-VSDO}) answers an $\alpha$-approximate distance from $s$ to $t$ in $G-x$ for any query $(x, t)$. It is a data structure version of the so-called single-source replacement path problem (SSRP). In this paper, we present a new \emph{nearly linear time} algorithm of constructing the $(1 + \epsilon)$-VSDO for any weighted directed graph of $n$ vertices and $m$ edges with integer weights in range $[1, W]$, and any positive constant $\epsilon \in (0, 1]$. More precisely, the presented oracle attains $\tilde{O}(m / \epsilon + n /\epsilon^2)$ construction time, $\tilde{O}(n/ \epsilon)$ size, and $\tilde{O}(1/\epsilon)$ query time for any polynomially-bounded $W$. To the best of our knowledge, this is the first non-trivial result for SSRP/VSDO beating the trivial $\tilde{O}(mn)$ computation time for directed graphs with polynomially-bounded edge weights. Such a result has been unknown so far even for the setting of $(1 + \epsilon)$-approximation. It also implies that the known barrier of $\Omega(m\sqrt{n})$ time for the exact SSRP by Chechik and Magen~[ICALP2020] does not apply to the case of approximation. | cs | 0 |
Title: Linguistic Profiling of Deepfakes: An Open Database for Next-Generation Deepfake Detection
Abstract: The emergence of text-to-image generative models has revolutionized the field of deepfakes, enabling the creation of realistic and convincing visual content directly from textual descriptions. However, this advancement presents considerably greater challenges in detecting the authenticity of such content. Existing deepfake detection datasets and methods often fall short in effectively capturing the extensive range of emerging deepfakes and offering satisfactory explanatory information for detection. To address the significant issue, this paper introduces a deepfake database (DFLIP-3K) for the development of convincing and explainable deepfake detection. It encompasses about 300K diverse deepfake samples from approximately 3K generative models, which boasts the largest number of deepfake models in the literature. Moreover, it collects around 190K linguistic footprints of these deepfakes. The two distinguished features enable DFLIP-3K to develop a benchmark that promotes progress in linguistic profiling of deepfakes, which includes three sub-tasks namely deepfake detection, model identification, and prompt prediction. The deepfake model and prompt are two essential components of each deepfake, and thus dissecting them linguistically allows for an invaluable exploration of trustworthy and interpretable evidence in deepfake detection, which we believe is the key for the next-generation deepfake detection. Furthermore, DFLIP-3K is envisioned as an open database that fosters transparency and encourages collaborative efforts to further enhance its growth. Our extensive experiments on the developed benchmark verify that our DFLIP-3K database is capable of serving as a standardized resource for evaluating and comparing linguistic-based deepfake detection, identification, and prompt prediction techniques. | cs | 0 |
Title: Atomistic modelling of near-crack-tip plasticity
Abstract: An atomistic model of near-crack-tip plasticity on a square lattice under anti-plane shear kinematics is formulated and studied. The model is based upon a new geometric and functional framework of a lattice manifold complex, which ensures that the crack surface is fully taken into account, while preserving the crucial notion of duality. As a result, existence of locally stable equilibrium configurations containing both a crack opening and dislocations is established. Notably, with the boundary in the form of a crack surface accounted for, no minimum separation between a dislocation core and the crack surface or the crack tip is required. The work presented here constitutes a foundation for several further studies aiming to put the phenomenon of near-crack-tip plasticity on a rigorous footing. | math | 1 |
Title: An Asymmetric Contrastive Loss for Handling Imbalanced Datasets
Abstract: Contrastive learning is a representation learning method performed by contrasting a sample to other similar samples so that they are brought closely together, forming clusters in the feature space. The learning process is typically conducted using a two-stage training architecture, and it utilizes the contrastive loss (CL) for its feature learning. Contrastive learning has been shown to be quite successful in handling imbalanced datasets, in which some classes are overrepresented while some others are underrepresented. However, previous studies have not specifically modified CL for imbalanced datasets. In this work, we introduce an asymmetric version of CL, referred to as ACL, in order to directly address the problem of class imbalance. In addition, we propose the asymmetric focal contrastive loss (AFCL) as a further generalization of both ACL and focal contrastive loss (FCL). Results on the FMNIST and ISIC 2018 imbalanced datasets show that AFCL is capable of outperforming CL and FCL in terms of both weighted and unweighted classification accuracies. In the appendix, we provide a full axiomatic treatment on entropy, along with complete proofs. | cs | 1 |
Title: Flip Graphs on Self-Complementary Ideals of Chain Products
Abstract: In this paper, we introduce a flip operation on self-complementary ideals of chain product posets and study the resulting flip graphs. We give asymptotics for the number of vertices in these graphs, compute their diameters, and give bounds for their radii. We also define similar flip operations on self-complementary ideals of the chain product $[2r]\times [2r]\times [2r]$ satisfying additional symmetries, and we achieve similar results for the resulting flip graphs. | math | 0 |
Title: Dynamic Service Placement for Mobile Micro-Clouds with Predicted Future Costs
Abstract: Mobile micro-clouds are promising for enabling performance-critical cloud applications. However, one challenge therein is the dynamics at the network edge. In this paper, we study how to place service instances to cope with these dynamics, where multiple users and service instances coexist in the system. Our goal is to find the optimal placement (configuration) of instances to minimize the average cost over time, leveraging the ability of predicting future cost parameters with known accuracy. We first propose an offline algorithm that solves for the optimal configuration in a specific look-ahead time-window. Then, we propose an online approximation algorithm with polynomial time-complexity to find the placement in real-time whenever an instance arrives. We analytically show that the online algorithm is $O(1)$-competitive for a broad family of cost functions. Afterwards, the impact of prediction errors is considered and a method for finding the optimal look-ahead window size is proposed, which minimizes an upper bound of the average actual cost. The effectiveness of the proposed approach is evaluated by simulations with both synthetic and real-world (San Francisco taxi) user-mobility traces. The theoretical methodology used in this paper can potentially be applied to a larger class of dynamic resource allocation problems. | cs | 1 |
Title: Kernel Search approach to solve the Minimum Spanning Tree Problem with conflicting edge pairs
Abstract: The Minimum Spanning Tree Problem with Conflicts consists in finding the minimum conflict-free spanning tree of a graph, i.e., the spanning tree of minimum cost, including no pairs of edges that are in conflict. In this paper, we solve this problem using a tailored Kernel Search heuristic method, which consists in solving iteratively improved restrictions of the problem. The main novelty of the approach consists in using an independent set of the conflict graph within the algorithm. We test our approach on the benchmark instances and we compare our results with the ones obtained by other heuristics available in the literature. | math | 0 |
Title: A Case Study on Software Vulnerability Coordination
Abstract: Context: Coordination is a fundamental tenet of software engineering. Coordination is required also for identifying discovered and disclosed software vulnerabilities with Common Vulnerabilities and Exposures (CVEs). Motivated by recent practical challenges, this paper examines the coordination of CVEs for open source projects through a public mailing list. Objective: The paper observes the historical time delays between the assignment of CVEs on a mailing list and the later appearance of these in the National Vulnerability Database (NVD). Drawing from research on software engineering coordination, software vulnerabilities, and bug tracking, the delays are modeled through three dimensions: social networks and communication practices, tracking infrastructures, and the technical characteristics of the CVEs coordinated. Method: Given a period between 2008 and 2016, a sample of over five thousand CVEs is used to model the delays with nearly fifty explanatory metrics. Regression analysis is used for the modeling. Results: The results show that the CVE coordination delays are affected by different abstractions for noise and prerequisite constraints. These abstractions convey effects from the social network and infrastructure dimensions. Particularly strong effect sizes are observed for annual and monthly control metrics, a control metric for weekends, the degrees of the nodes in the CVE coordination networks, and the number of references given in NVD for the CVEs archived. Smaller but visible effects are present for metrics measuring the entropy of the emails exchanged, traces to bug tracking systems, and other related aspects. The empirical signals are weaker for the technical characteristics. Conclusion: [...] | cs | 1 |
Title: Comparative Analysis of Obstacle Approximation Strategies for the Steady Incompressible Navier-Stokes Equations
Abstract: This paper aims to compare and evaluate various obstacle approximation techniques employed in the context of the steady incompressible Navier-Stokes equations. Specifically, we investigate the effectiveness of a standard volume penalization approximation and an approximation method utilizing high viscosity inside the obstacle region, as well as their composition. Analytical results concerning the convergence rate of these approaches are provided, and extensive numerical experiments are conducted to validate their performance. | math | 0 |
Title: Notes on Nonrepetitive Graph Colouring
Abstract: A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree $\Delta$ has a $f(\Delta)$-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree $\Delta$ has a $O(k\Delta)$-colouring that is nonrepetitive on paths, and a $O(k\Delta^3)$-colouring that is nonrepetitive on walks. | math | 1 |
Title: The continuum disordered pinning model
Abstract: Any renewal processes on $\mathbb{N}$ with a polynomial tail, with exponent $\alpha \in (0,1)$, has a non-trivial scaling limit, known as the $\alpha$-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for $\alpha \in (1/2, 1)$ these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of $\mathbb{R}$ in a white noise random environment, with subtle features: -Any fixed a.s. property of the $\alpha$-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment. -Nonetheless, the law of the CDPM is singular with respect to the law of the $\alpha$-stable regenerative set, for almost every realization of the environment. The existence of a disordered continuum model, such as the CDPM, is a manifestation of disorder relevance for pinning models with $\alpha \in (1/2, 1)$. | math | 1 |
Title: On some anabelian properties of arithmetic curves
Abstract: In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $\pi_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary points inside it, the $p$-part of the cyclotomic character, the number of points on the boundary of all finite etale covers, etc. Under certain finiteness hypothesis on Tate-Shafarevich groups with divisible coefficients, one can reconstruct all this quantities from the fundamental group alone. | math | 1 |
Title: Parallel Algorithms Align with Neural Execution
Abstract: Neural algorithmic reasoners are parallel processors. Teaching them sequential algorithms contradicts this nature, rendering a significant share of their computations redundant. Parallel algorithms however may exploit their full computational power, therefore requiring fewer layers to be executed. This drastically reduces training times, as we observe when comparing parallel implementations of searching, sorting and finding strongly connected components to their sequential counterparts on the CLRS framework. Additionally, parallel versions achieve (often strongly) superior predictive performance. | cs | 0 |
Title: Hyperbolicity and non-conservativity of a hydrodynamic model of swarming rigid bodies
Abstract: In this paper, we study a nonlinear system of first order partial differential equations describing the macroscopic behavior of an ensemble of interacting self-propelled rigid bodies. Such system may be relevant for the modelling of bird flocks, fish schools or fleets of drones. We show that the system is hyperbolic and can be approximated by a conservative system through relaxation. We also derive viscous corrections to the model from the hydrodynamic limit of a kinetic model. This analysis prepares the future development of numerical approximations of this system. | math | 1 |
Title: The Rotation of Eigenspaces of Perturbed Matrix Pairs
Abstract: We revisit the relative perturbation theory for invariant subspaces of positive definite matrix pairs. As a prototype model problem for our results we consider parameter dependent families of eigenvalue problems. We show that new estimates are a natural way to obtain sharp --- as functions of the parameter indexing the family of matrix pairs --- estimates for the rotation of spectral subspaces. | math | 1 |
Title: CURE Dataset: Ladder Networks for Audio Event Classification
Abstract: Audio event classification is an important task for several applications such as surveillance, audio, video and multimedia retrieval etc. There are approximately 3M people with hearing loss who can't perceive events happening around them. This paper establishes the CURE dataset which contains curated set of specific audio events most relevant for people with hearing loss. We propose a ladder network based audio event classifier that utilizes 5s sound recordings derived from the Freesound project. We adopted the state-of-the-art convolutional neural network (CNN) embeddings as audio features for this task. We also investigate extreme learning machine (ELM) for event classification. In this study, proposed classifiers are compared with support vector machine (SVM) baseline. We propose signal and feature normalization that aims to reduce the mismatch between different recordings scenarios. Firstly, CNN is trained on weakly labeled Audioset data. Next, the pre-trained model is adopted as feature extractor for proposed CURE corpus. We incorporate ESC-50 dataset as second evaluation set. Results and discussions validate the superiority of Ladder network over ELM and SVM classifier in terms of robustness and increased classification accuracy. While Ladder network is robust to data mismatches, simpler SVM and ELM classifiers are sensitive to such mismatches, where the proposed normalization techniques can play an important role. Experimental studies with ESC-50 and CURE corpora elucidate the differences in dataset complexity and robustness offered by proposed approaches. | cs | 1 |
Title: On the Girth of Graph Lifts
Abstract: The size of the smallest $k$-regular graph of girth $g$ is denoted by the well studied function $n(k,g)$. We suggest generalizing this function to $n(H,g)$, defined as the smallest size girth $g$ graph covering the, possibly non-regular, graph $H$. We prove that the two main combinatorial bounds on $n(k,g)$, the Moore lower bound and the Erd\"{o}s Sachs upper bound, carry over to the new setting of lifts, even in their non-asymptotic form. We also consider two other generalizations of $n(k,g)$: i) The smallest size girth $g$ graph sharing a universal cover with $H$. We prove that it is the same as $n(H,g)$ up to a multiplicative constant. ii) The smallest size girth $g$ graph with a prescribed degree distribution. We discuss this known generalization and argue that the new suggested definitions are superior. We conclude with experimental results for a specific base graph and with some conjectures and open problems. | math | 0 |
Title: A survey on divisibility of ultrafilters
Abstract: An extension of the divisibility relation on $\mathbb{N}$ to the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding a few simple results connecting the themes of different stages of the research. The main highlights include: separation into the lower part $L$ (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter; estimating cardinalities of divisibility-equivalence classes; extending the congruence relation (in two ways) and checking properties of the obtained relations. | math | 0 |
Title: Simple loops on 2-bridge spheres in Heckoid orbifolds for the trivial knot
Abstract: In this paper, we give a necessary and sufficient condition for an essential simple loop on a $2$-bridge sphere in an even Heckoid orbifold for the trivial knot to be null-homotopic, peripheral or torsion in the orbifold. We also give a necessary and sufficient condition for two essential simple loops on a $2$-bridge sphere in an even Heckoid orbifold for the trivial knot to be homotopic in the orbifold. | math | 1 |
Title: Asymptotic probability for connectedness
Abstract: We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of other derivative combinatorial classes. The general result applies to rapidly growing combinatorial structures, which we call gargantuan, that also admit a sequence decomposition. The result is then applied to several models of graphs, of surfaces (square-tiled surfaces, combinatorial maps), and to geometric models of higher dimension (constellations, graph encoded manifolds). The corresponding derivative combinatorial classes are irreducible (multi)tournaments, indecomposable (multi)permutations and indecomposable perfect (multi)matchings. | math | 0 |
Title: Strong asymptotic expansions in a multidirection
Abstract: In this paper we prove that, for asymptotically bounded holomorphic functions defined in a polysector in ${\mathbb C}^n$, the existence of a strong asymptotic expansion in Majima's sense following a single multidirection towards the vertex entails (global) asymptotic expansion in the whole polysector. Moreover, we specialize this result for Gevrey strong asymptotic expansions. This is a generalization of a result proved by A. Fruchard and C. Zhang for asymptotic expansions in one variable, but the proof, mainly in the Gevrey case, involves different techniques of a functional-analytic nature. | math | 1 |
Title: Quadratic Discontinuous Galerkin methods for Unilateral Contact Problem
Abstract: In this article, we employ discontinuous Galerkin (DG) methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first establish an optimal \textit{a priori} error estimates under the appropriate regularity assumption on the exact solution $\b{u}$. Further, we analyze \textit{a posteriori} error estimates in the DG norm wherein, the reliability and efficiency of the proposed \textit{a posteriori} error estimator is addressed. The suitable construction of discrete Lagrange multiplier $\b{\lambda_h}$ and some intermediate operators play a key role in developing \textit{a posteriori} error analysis. Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings. | math | 0 |
Title: TinyLlama: An Open-Source Small Language Model
Abstract: We present TinyLlama, a compact 1.1B language model pretrained on around 1 trillion tokens for approximately 3 epochs. Building on the architecture and tokenizer of Llama 2, TinyLlama leverages various advances contributed by the open-source community (e.g., FlashAttention), achieving better computational efficiency. Despite its relatively small size, TinyLlama demonstrates remarkable performance in a series of downstream tasks. It significantly outperforms existing open-source language models with comparable sizes. Our model checkpoints and code are publicly available on GitHub at https://github.com/jzhang38/TinyLlama. | cs | 0 |
Title: On the Performance of Large-Scale Wireless Networks in the Finite Block-Length Regime
Abstract: Ultra-Reliable Low-Latency Communications have stringent delay constraints, and hence use codes with small block length (short codewords). In these cases, classical models that provide good approximations to systems with infinitely long codewords become imprecise. To remedy this, in this paper, an average coding rate expression is derived for a large scale network with short codewords using stochastic geometry and the theory of coding in the finite blocklength regime. The average coding rate and upper and lower bounds on the outage probability of the large-scale network are derived, and a tight approximation of the outage probability is presented. Then, simulations are presented to study the effect of network parameters on the average coding rate and the outage probability of the network, which demonstrate that results in the literature derived for the infinite blocklength regime overestimate the network performance, whereas the results in this paper provide a more realistic performance evaluation. | cs | 1 |
Title: Flip colouring of graphs II
Abstract: We give results concerning two problems on the recently introduced flip colourings of graphs, a new class of local v. global phenomena. We prove that for $(b, r)$-flip sequences with $4 \leq b < r < b + 2 \left\lfloor\frac{b+2}{6}\right\rfloor^2$, small constructions of $(b,r)$-flip graphs on $O(b+r)$ vertices are possible. Furthermore, we prove that there exists $k$-flip sequences $(a_1, \dots, a_k)$ where $k > 4$, such that $a_k$ can be arbitrarily large whilst $a_i$ is constant for $1 \leq i < \frac{k}{4}$. | math | 0 |
Title: Hessian of the Ricci Calabi functional
Abstract: The Ricci Calabi functional is a functional on the space of K\"ahler metrics of Fano manifolds. Its critical points are called generalized K\"ahler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized K\"ahler Einstein metrics. As its application, we give another proof of a Matsushima's type decomposition theorem for holomorphic vector fields, which was originally proved by Mabuchi. We also discuss a relation to the inverse Monge-Amp\`ere flow developed recently by Collins-Hisamoto-Takahashi. | math | 1 |
Title: Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces
Abstract: Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek-Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an "inflating" sphere through a point $p$ in a Damek-Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $\gamma$ starting from $p$, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by "overinflating the horospheres" by pushing the center of the sphere beyond the point at infinity of $\gamma$ along a virtual prolongation of $\gamma$. They give a new family of isoparametric hypersurfaces in Damek-Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. D\'iaz-Ramos and M. Dom\'inguez-V\'azquez [arXiv:1111.0264] in Damek-Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties. | math | 0 |
Title: Simpler Specifications and Easier Proofs of Distributed Algorithms Using History Variables
Abstract: This paper studies specifications and proofs of distributed algorithms when only message history variables are used, using the Basic Paxos and Multi-Paxos algorithms for distributed consensus as precise case studies. We show that not using and maintaining other state variables yields simpler specifications that are more declarative and easier to understand. It also allows easier proofs to be developed by needing fewer invariants and facilitating proof derivations. Furthermore, the proofs are mechanically checked more efficiently. We show that specifications in TLA+, Lamport's temporal logic of actions, and proofs in TLAPS, the TLA+ Proof System (TLAPS) are reduced by a quarter or more for single-value Paxos and by about half or more for multi-value Paxos. Overall we need about half as many manually written invariants and proof obligations. Our proof for Basic Paxos takes about 25% less time for TLAPS to check, and our proofs for Multi-Paxos are checked within 1.5 minutes whereas prior proofs fail to be checked by TLAPS. | cs | 1 |
Title: A reciprocity relation for the twisted second moment of the Riemman Zeta function
Abstract: We prove a reciprocity relation for the twisted second moment of the Riemann Zeta function. This provides an analogue to a formula of Conrey for Dirichlet L-functions | math | 0 |
Title: Applications of machine learning and IoT for Outdoor Air Pollution Monitoring and Prediction: A Systematic Literature Review
Abstract: According to the World Health Organization (WHO), air pollution kills seven million people every year. Outdoor air pollution is a major environmental health problem affecting low, middle, and high-income countries. In the past few years, the research community has explored IoT-enabled machine learning applications for outdoor air pollution prediction. The general objective of this paper is to systematically review applications of machine learning and Internet of Things (IoT) for outdoor air pollution prediction and the combination of monitoring sensors and input features used. Two research questions were formulated for this review. 1086 publications were collected in the initial PRISMA stage. After the screening and eligibility phases, 37 papers were selected for inclusion. A cost-based analysis was conducted on the findings to highlight high-cost monitoring, low-cost IoT and hybrid enabled prediction. Three methods of prediction were identified: time series, feature-based and spatio-temporal. This review's findings identify major limitations in applications found in the literature, namely lack of coverage, lack of diversity of data and lack of inclusion of context-specific features. This review proposes directions for future research and underlines practical implications in healthcare, urban planning, global synergy and smart cities. | cs | 0 |
Title: Mobile ALOHA: Learning Bimanual Mobile Manipulation with Low-Cost Whole-Body Teleoperation
Abstract: Imitation learning from human demonstrations has shown impressive performance in robotics. However, most results focus on table-top manipulation, lacking the mobility and dexterity necessary for generally useful tasks. In this work, we develop a system for imitating mobile manipulation tasks that are bimanual and require whole-body control. We first present Mobile ALOHA, a low-cost and whole-body teleoperation system for data collection. It augments the ALOHA system with a mobile base, and a whole-body teleoperation interface. Using data collected with Mobile ALOHA, we then perform supervised behavior cloning and find that co-training with existing static ALOHA datasets boosts performance on mobile manipulation tasks. With 50 demonstrations for each task, co-training can increase success rates by up to 90%, allowing Mobile ALOHA to autonomously complete complex mobile manipulation tasks such as sauteing and serving a piece of shrimp, opening a two-door wall cabinet to store heavy cooking pots, calling and entering an elevator, and lightly rinsing a used pan using a kitchen faucet. Project website: https://mobile-aloha.github.io | cs | 0 |
Title: Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras
Abstract: The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $H$ is a CQG algebra. | math | 1 |