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Title: On friendship and cyclic parking functions
Abstract: In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars have the same preferred spot. In classical parking functions, if a car's preferred spot is already occupied by a previous car, it drives forward and looks for the first unoccupied spot to park. In this work, we introduce a variant of classical parking functions, called "friendship parking functions", which imposes additional restrictions on where cars can park. Namely, a car can only end up parking next to cars which are its friends (friendship will correspond to adjacency in an underlying graph). We characterise and enumerate such friendship parking functions according to their outcome permutation, which describes the final configuration when all cars have parked. We apply this to the case where the underlying friendship graph is the cycle graph. Finally, we consider a subset of classical parking functions, called "cyclic parking functions", where cars end up in an increasing cyclic order. We enumerate these cyclic parking functions and exhibit a bijection to permutation components. | math | 0 |
Title: Towards Unsupervised Open World Semantic Segmentation
Abstract: For the semantic segmentation of images, state-of-the-art deep neural networks (DNNs) achieve high segmentation accuracy if that task is restricted to a closed set of classes. However, as of now DNNs have limited ability to operate in an open world, where they are tasked to identify pixels belonging to unknown objects and eventually to learn novel classes, incrementally. Humans have the capability to say: I don't know what that is, but I've already seen something like that. Therefore, it is desirable to perform such an incremental learning task in an unsupervised fashion. We introduce a method where unknown objects are clustered based on visual similarity. Those clusters are utilized to define new classes and serve as training data for unsupervised incremental learning. More precisely, the connected components of a predicted semantic segmentation are assessed by a segmentation quality estimate. connected components with a low estimated prediction quality are candidates for a subsequent clustering. Additionally, the component-wise quality assessment allows for obtaining predicted segmentation masks for the image regions potentially containing unknown objects. The respective pixels of such masks are pseudo-labeled and afterwards used for re-training the DNN, i.e., without the use of ground truth generated by humans. In our experiments we demonstrate that, without access to ground truth and even with few data, a DNN's class space can be extended by a novel class, achieving considerable segmentation accuracy. | cs | 1 |
Title: A dimension reduction approach for loss valuation in credit risk modelling
Abstract: This paper addresses the ``curse of dimensionality'' in the loss valuation of credit risk models. A dimension reduction methodology based on the Bayesian filter and smoother is proposed. This methodology is designed to achieve a fast and accurate loss valuation algorithm in credit risk modelling, but it can also be extended to valuation models of other risk types. The proposed methodology is generic, robust and can easily be implemented. Moreover, the accuracy of the proposed methodology in the estimation of expected loss and value-at-risk is illustrated by numerical experiments. The results suggest that, compared to the currently most used PCA approach, the proposed methodology provides more accurate estimation of expected loss and value-at-risk of a loss distribution. | cs | 0 |
Title: Machine Learning in Robotic Ultrasound Imaging: Challenges and Perspectives
Abstract: This article reviews the recent advances in intelligent robotic ultrasound (US) imaging systems. We commence by presenting the commonly employed robotic mechanisms and control techniques in robotic US imaging, along with their clinical applications. Subsequently, we focus on the deployment of machine learning techniques in the development of robotic sonographers, emphasizing crucial developments aimed at enhancing the intelligence of these systems. The methods for achieving autonomous action reasoning are categorized into two sets of approaches: those relying on implicit environmental data interpretation and those using explicit interpretation. Throughout this exploration, we also discuss practical challenges, including those related to the scarcity of medical data, the need for a deeper understanding of the physical aspects involved, and effective data representation approaches. Moreover, we conclude by highlighting the open problems in the field and analyzing different possible perspectives on how the community could move forward in this research area. | cs | 0 |
Title: Betting strategies with bounded splits
Abstract: We show that a pair of Kolmogorov-Loveland betting strategies cannot win on every non-Martin-L\"of random sequence if either of the two following conditions is true: (I) There is an unbounded computable function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\ell$, bet on at most $\ell-g(\ell)$ positions among the first $\ell$ positions of the sequence. (II) There is a sublinear function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\ell$, bet on at least $\ell-g(\ell)$ positions among the first $\ell$ positions of the sequence. | cs | 1 |
Title: Nilpotent polynomials over $\mathbb{Z}$
Abstract: For a polynomial $u(x)$ in $\mathbb{Z}[x]$ and $r\in\mathbb{Z}$, we consider the orbit of $u$ at $r$ denoted and defined by $\mathcal{O}_u(r):=\{u^{(n)}(r)~|~n\in\mathbb{N}\}$. Here we study polynomials for which $0$ is in the orbit for a given $r$. We provide a complete classification of these polynomials when $|r|\le 4$, with $|r|\le 1$ already done in \cite{SS23}. The central goal of this paper is to study the following questions: (i) relationship between the integers $r$ and $m$, for a polynomial $u$ in $N_{r,m}$; (ii) classification of the polynomials with nilpotency index $|r|$ for large enough $|r|$; and (iii) integer sequences with a generating polynomial. | math | 0 |
Title: Fading memory as inductive bias in residual recurrent networks
Abstract: Residual connections have been proposed as an architecture-based inductive bias to mitigate the problem of exploding and vanishing gradients and increased task performance in both feed-forward and recurrent networks (RNNs) when trained with the backpropagation algorithm. Yet, little is known about how residual connections in RNNs influence their dynamics and fading memory properties. Here, we introduce weakly coupled residual recurrent networks (WCRNNs) in which residual connections result in well-defined Lyapunov exponents and allow for studying properties of fading memory. We investigate how the residual connections of WCRNNs influence their performance, network dynamics, and memory properties on a set of benchmark tasks. We show that several distinct forms of residual connections yield effective inductive biases that result in increased network expressivity. In particular, those are residual connections that (i) result in network dynamics at the proximity of the edge of chaos, (ii) allow networks to capitalize on characteristic spectral properties of the data, and (iii) result in heterogeneous memory properties. In addition, we demonstrate how our results can be extended to non-linear residuals and introduce a weakly coupled residual initialization scheme that can be used for Elman RNNs. | cs | 0 |
Title: Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots
Abstract: In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}. | cs | 1 |
Title: Phototactic bioconvection in an algal suspension with a free top wall due to diffuse flux in the absence of direct collimated flux
Abstract: In this article, the effect of diffuse flux in the absence of direct collimated flux on the onset of phototactic bioconvection is investigated. The main effect of diffuse flux in the absence of collimated flux is on the swimming behaviour of microorganisms in the suspension. At higher diffuse flux, the horizontal component of swimming orientation exhibits a higher magnitude, which slows the rate of pattern formation in the suspension. Also, the linear stability of the suspension predicts that the most unstable mode of disturbance transits from stationary (oscillatory) to oscillatory (stationary) at the variation in the magnitude of diffuse flux for some fixed parameters. The overstable nature of disturbance is mostly observed at the high value of swimming speed and extinction coefficient. Moreover, the suspension shows a more stable behaviour at the higher magnitude of diffuse flux. | math | 0 |
Title: Whittaker modules and hyperbolic Toda lattices
Abstract: Let $\sg$ be a complex finite-dimensional simple Lie algebra and let $\sg_l$ be the corresponding generalized Takiff algebra. This paper studies the affine variety $\ssf+\sb_l$ where $\ssf$ is similar to a principal nilpotent element of $\sg$ and $\sb_l$ is a subalgebra corresponding to the Borel subalgebra $\sb$ of $\sg$. Inspired by Kostant's work then we deal with two questions. One of them is to construct the Whittaker model for the $G_l$-invariants of symmetric algebra $S(\sg_l)$ where $G_l$ is the adjoint group of $\sg_l$ and $G_l$ acts on $S(\sg_l)$ by coadjoint action, and then to classify all nonsingular Whittaker modules over $\sg_l$. Another one is to describe the symplectic structure of the manifold $Z\subseteq\ssf+\sb_l$ of normalized Jacobi elements. Then the Hamiltonian corresponding to a fundamental invariant provides a class of hyperbolic Toda lattices. In particular, a simplest example describes the state of a dynamical system consisting of a positive mass particle and a negative mass particle. | math | 0 |
Title: On Memorization and Privacy Risks of Sharpness Aware Minimization
Abstract: In many recent works, there is an increased focus on designing algorithms that seek flatter optima for neural network loss optimization as there is empirical evidence that it leads to better generalization performance in many datasets. In this work, we dissect these performance gains through the lens of data memorization in overparameterized models. We define a new metric that helps us identify which data points specifically do algorithms seeking flatter optima do better when compared to vanilla SGD. We find that the generalization gains achieved by Sharpness Aware Minimization (SAM) are particularly pronounced for atypical data points, which necessitate memorization. This insight helps us unearth higher privacy risks associated with SAM, which we verify through exhaustive empirical evaluations. Finally, we propose mitigation strategies to achieve a more desirable accuracy vs privacy tradeoff. | cs | 0 |
Title: A leaf as a convex domain containing eigenvalues of a linear transformation on a real inner product space
Abstract: We define a leaf which is a domain in the closure $\overline{\mathfrak{H}}=\mathfrak{H}\cup\mathbb{R}\cup\{\infty\}$ of the complex upper half plane $\mathfrak{H} = \{z\in\mathbb{C}\mid{\mathrm{Im}\,} z>0\}$ for any linear transformation of an inner product space over the real number field $\mathbb{R}$. If the dimension of the inner product space is at least 3, the leaf is convex on the Poincar\'e metric, and then simply connected, and contains all eigenvalues with nonnegative imaginary part. Moreover, if the linear transformation is normal, the leaf is the eigenvalue geodesic polygon, which is the minimum convex domain in $\overline{\mathfrak{H}}$ containing all eigenvalues with nonnegative imaginary part. We discuss the application of the geometric properties of a leaf to the operator norm. | math | 0 |
Title: Federated Calibration and Evaluation of Binary Classifiers
Abstract: We address two major obstacles to practical use of supervised classifiers on distributed private data. Whether a classifier was trained by a federation of cooperating clients or trained centrally out of distribution, (1) the output scores must be calibrated, and (2) performance metrics must be evaluated -- all without assembling labels in one place. In particular, we show how to perform calibration and compute precision, recall, accuracy and ROC-AUC in the federated setting under three privacy models (i) secure aggregation, (ii) distributed differential privacy, (iii) local differential privacy. Our theorems and experiments clarify tradeoffs between privacy, accuracy, and data efficiency. They also help decide whether a given application has sufficient data to support federated calibration and evaluation. | cs | 1 |
Title: Some remarks on infinitesimals in MV-algebras
Abstract: Replacing $\{0\}$ by the whole ideal of infinitesimals yields a weaker notion of \emph{archimedean element} that we call \emph{quasiarchimedean}. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being \emph{quasihyperarchimedean} \mbox{MV-algebras,} which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MV-algebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the \mbox{set-theoretical} usually found in the literature. | math | 1 |
Title: Towards Safe and Collaborative Robotic Ultrasound Tissue Scanning in Neurosurgery
Abstract: Intraoperative ultrasound imaging is used to facilitate safe brain tumour resection. However, due to challenges with image interpretation and the physical scanning, this tool has yet to achieve widespread adoption in neurosurgery. In this paper, we introduce the components and workflow of a novel, versatile robotic platform for intraoperative ultrasound tissue scanning in neurosurgery. An RGB-D camera attached to the robotic arm allows for automatic object localisation with ArUco markers, and 3D surface reconstruction as a triangular mesh using the ImFusion Suite software solution. Impedance controlled guidance of the US probe along arbitrary surfaces, represented as a mesh, enables collaborative US scanning, i.e., autonomous, teleoperated and hands-on guided data acquisition. A preliminary experiment evaluates the suitability of the conceptual workflow and system components for probe landing on a custom-made soft-tissue phantom. Further assessment in future experiments will be necessary to prove the effectiveness of the presented platform. | cs | 0 |
Title: Abacus models for parabolic quotients of affine Weyl groups
Abstract: We introduce abacus diagrams that describe minimal length coset representatives in affine Weyl groups of types B, C, and D. These abacus diagrams use a realization of the affine Weyl group of type C due to Eriksson to generalize a construction of James for the symmetric group. We also describe several combinatorial models for these parabolic quotients that generalize classical results in affine type A related to core partitions. | math | 1 |
Title: Eulerian 2-Complexes
Abstract: It is shown that Euler's theorem for graphs can be generalized for 2-complexes. Two notions that generalize cycle and Eulerian tour are introduced (``circlet'' and ``Eulerian cover''), and we show that for a strongly-connected, pure 2-complex, the following are equivalent: (i) each edge meets a positive even number of 2-cells (faces), (ii) the complex can be decomposed as the face-disjoint union of circlets, and (iii) the complex has an Eulerian cover. A number of examples are provided. | math | 0 |
Title: Unified Descriptive Intensional Logic
Abstract: UDIL (unified descriptive intensional logic) aims to be an alternative and improved version of Bealer's logic fulfilling the goal of unifying Bealer's systems T1 and T2 together with adding features to deal with definite descriptions and singular terms and their related philosophical problems (there are interesting connections to Zalta's more restricted parallel second-order version in his book \emph{Axiomatic Metaphysics}). UDIL also allows a much shorter and transparent proof of soundness, in particular with regards to a notoriously difficult preliminary lemma. UDIL stands out as being both formally and philosophically distinct from mainstream approaches to intensionality. One motivation for UDIL is to contribute to the Leibnizean goal of a formal philosophy, that is, a philosophy in which arguments and proofs are carried out entirely within a formal system. | math | 0 |
Title: The singular limit of the Water-Waves equations in the rigid lid regime
Abstract: The re-scaled Water-Waves equations depend strongly on the ratio epsilon between the amplitude of the wave and the depth of the water. We investigate in this paper the convergence as epsilon goes to zero of the free surface Euler equations to the so called rigid lid model. We first prove that the only solutions of this model are zero. Due to the conservation of the Hamiltonian, the solutions of the free surface Euler equations converge weakly to zero, but not strongly in the general case, as epsilon goes to zero. We then study this default of convergence. More precisely, we show a strong convergence result of the solutions of the water waves equations in the Zakharov-Craig-Sulem formulation to the solutions of the linear water-waves equations. It is then easy to observe that these latter converge weakly to zero. The simple structure of this system also allows us to explain the mechanisms of the weak convergence to zero. Finally, we show that this convergence to the rigid lid model also holds for the solutions of the Euler equations. To this end we give a new proof of the equivalence of the free surface Euler equations and of the Zakharov-Craig-Sulem equation by building an extension of the velocity and pressure fields. | math | 1 |
Title: Quota Trees
Abstract: We introduce the notion of quota trees in directed graphs. Given a nonnegative integer ``quota'' for each vertex of a directed multigraph $G$, a quota tree is an immersed rooted tree which hits each vertex of $G$ the prescribed number of times. When the quotas are all one, the tree is actually embedded and we recover the usual notion of a spanning arborescence (directed spanning tree). The usual algorithms which produce spanning arborescences with various properties typically have (sometimes more complicated) ``quota'' analogues. Our original motivation for studying quota trees was the problem of characterizing the sizes of the Myhill-Nerode equivalence classes in a connected deterministic finite-state automaton recognizing a given regular language. We show that the obstruction to realizing a given set of M-N class sizes is precisely the existence of a suitable quota tree. In this paper we develop the basic theory of quota trees. We give necessary and sufficient conditions for the existence of a quota tree (or forest) over a given directed graph with specified quotas, solving the M-N class size problem as a special case. We discuss some potential applications of quota trees and forests, and connect them to the $k$ lightest paths problem. We give two proofs of the main theorem: one based on an algorithmic loop invariant, and one based on direct enumeration of quota trees. For the latter, we use Lagrange inversion to derive a formula which vastly generalizes both the matrix-tree theorem and Cayley's formula for counting labeled trees. We give an efficient algorithm to sample uniformly from the set of forests with given quotas, as well as a generalization of Edmonds' algorithm for computing a minimum-weight quota forest. | math | 0 |
Title: Method for Solving State-Path Constrained Optimal Control Problems Using Adaptive Radau Collocation
Abstract: A new method is developed for accurately approximating the solution to state-variable inequality path constrained optimal control problems using a multiple-domain adaptive Legendre-Gauss-Radau collocation method. The method consists of the following parts. First, a structure detection method is developed to estimate switch times in the activation and deactivation of state-variable inequality path constraints. Second, using the detected structure, the domain is partitioned into multiple-domains where each domain corresponds to either a constrained or an unconstrained segment. Furthermore, additional decision variables are introduced in the multiple-domain formulation, where these additional decision variables represent the switch times of the detected active state-variable inequality path constraints. Within a constrained domain, the path constraint is differentiated with respect to the independent variable until the control appears explicitly, and this derivative is set to zero along the constrained arc while all preceding derivatives are set to zero at the start of the constrained arc. The time derivatives of the active state-variable inequality path constraints are computed using automatic differentiation and the properties of the chain rule. The method is demonstrated on two problems, the first being a benchmark optimal control problem which has a known analytical solution and the second being a challenging problem from the field of aerospace engineering in which there is no known analytical solution. When compared against previously developed adaptive Legendre-Gauss-Radau methods, the results show that the method developed in this paper is capable of computing accurate solutions to problems whose solution contain active state-variable inequality path constraints. | math | 0 |
Title: Disjointness for measurably distal group actions and applications
Abstract: We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener--Wintner type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence. | math | 1 |
Title: Sobolev spaces on hypergroups Gelfand pairs
Abstract: This paper introduces Sobolev spaces over Gelfand pairs in the framework of hypergroups. The Sobolev spaces in question are constructed from the Fourier transform on hypergroup Gelfand pairs. Mainly, the paper focuses on the investigation of Sobolev embedding results. | math | 0 |
Title: From Language to Programs: Bridging Reinforcement Learning and Maximum Marginal Likelihood
Abstract: Our goal is to learn a semantic parser that maps natural language utterances into executable programs when only indirect supervision is available: examples are labeled with the correct execution result, but not the program itself. Consequently, we must search the space of programs for those that output the correct result, while not being misled by spurious programs: incorrect programs that coincidentally output the correct result. We connect two common learning paradigms, reinforcement learning (RL) and maximum marginal likelihood (MML), and then present a new learning algorithm that combines the strengths of both. The new algorithm guards against spurious programs by combining the systematic search traditionally employed in MML with the randomized exploration of RL, and by updating parameters such that probability is spread more evenly across consistent programs. We apply our learning algorithm to a new neural semantic parser and show significant gains over existing state-of-the-art results on a recent context-dependent semantic parsing task. | cs | 1 |
Title: Cyclic codes from low differentially uniform functions
Abstract: Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles [Discrete Math. 321, 2014] and [SIAM J. Discrete Math. 27(4), 2013], Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall provide answers to some open problems presented in the literature. The first one concerns Open Problem 1, proposed by Ding and Zhou in Discrete Math. 321, 2014. The two others are Open Problems 5.16 and 5.25, raised by Ding in [SIAM J. Discrete Math. 27(4), 2013]. | cs | 1 |
Title: Strongly minimal Steiner Systems III: Path graphs and sparse configurations
Abstract: We introduce a uniform method of proof for the following results. For {\em each} of the following conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying that condition: i) Theorem~2.2.4: (extending \cite{Chicoetal}) each Steiner triple system is $\infty$-sparse and has a uniform but not perfect path graph; ii) (Theorem~5.4.2: (extending \cite{CameronWebb}) each Steiner $k$-system (for $k=p^n$) is $2$-transitive and has a uniform path graph (infinite cycles only); iii) Theorem~2.1.5: (extending \cite{Fujiwaramitre}, each is anti-Pasch (anti-mitre); iv) Theorem~3.6 has an explicit quasi-group structure. In each case all members of the family satisfy the same complete strongly minimal theory and it has $\aleph_0$ countable models and one model of each uncountable cardinal. | math | 1 |
Title: Pixel personality for dense object tracking in a 2D honeybee hive
Abstract: Tracking large numbers of densely-arranged, interacting objects is challenging due to occlusions and the resulting complexity of possible trajectory combinations, as well as the sparsity of relevant, labeled datasets. Here we describe a novel technique of collective tracking in the model environment of a 2D honeybee hive in which sample colonies consist of $N\sim10^3$ highly similar individuals, tightly packed, and in rapid, irregular motion. Such a system offers universal challenges for multi-object tracking, while being conveniently accessible for image recording. We first apply an accurate, segmentation-based object detection method to build initial short trajectory segments by matching object configurations based on class, position and orientation. We then join these tracks into full single object trajectories by creating an object recognition model which is adaptively trained to recognize honeybee individuals through their visual appearance across multiple frames, an attribute we denote as pixel personality. Overall, we reconstruct ~46% of the trajectories in 5 min recordings from two different hives and over 71% of the tracks for at least 2 min. We provide validated trajectories spanning 3000 video frames of 876 unmarked moving bees in two distinct colonies in different locations and filmed with different pixel resolutions, which we expect to be useful in the further development of general-purpose tracking solutions. | cs | 1 |
Title: UpFusion: Novel View Diffusion from Unposed Sparse View Observations
Abstract: We propose UpFusion, a system that can perform novel view synthesis and infer 3D representations for an object given a sparse set of reference images without corresponding pose information. Current sparse-view 3D inference methods typically rely on camera poses to geometrically aggregate information from input views, but are not robust in-the-wild when such information is unavailable/inaccurate. In contrast, UpFusion sidesteps this requirement by learning to implicitly leverage the available images as context in a conditional generative model for synthesizing novel views. We incorporate two complementary forms of conditioning into diffusion models for leveraging the input views: a) via inferring query-view aligned features using a scene-level transformer, b) via intermediate attentional layers that can directly observe the input image tokens. We show that this mechanism allows generating high-fidelity novel views while improving the synthesis quality given additional (unposed) images. We evaluate our approach on the Co3Dv2 and Google Scanned Objects datasets and demonstrate the benefits of our method over pose-reliant sparse-view methods as well as single-view methods that cannot leverage additional views. Finally, we also show that our learned model can generalize beyond the training categories and even allow reconstruction from self-captured images of generic objects in-the-wild. | cs | 0 |
Title: Tailor: Size Recommendations for High-End Fashion Marketplaces
Abstract: In the ever-changing and dynamic realm of high-end fashion marketplaces, providing accurate and personalized size recommendations has become a critical aspect. Meeting customer expectations in this regard is not only crucial for ensuring their satisfaction but also plays a pivotal role in driving customer retention, which is a key metric for the success of any fashion retailer. We propose a novel sequence classification approach to address this problem, integrating implicit (Add2Bag) and explicit (ReturnReason) user signals. Our approach comprises two distinct models: one employs LSTMs to encode the user signals, while the other leverages an Attention mechanism. Our best model outperforms SFNet, improving accuracy by 45.7%. By using Add2Bag interactions we increase the user coverage by 24.5% when compared with only using Orders. Moreover, we evaluate the models' usability in real-time recommendation scenarios by conducting experiments to measure their latency performance. | cs | 0 |
Title: Parsing using a grammar of word association vectors
Abstract: This paper was was first drafted in 2001 as a formalization of the system described in U.S. patent U.S. 7,392,174. It describes a system for implementing a parser based on a kind of cross-product over vectors of contextually similar words. It is being published now in response to nascent interest in vector combination models of syntax and semantics. The method used aggressive substitution of contextually similar words and word groups to enable product vectors to stay in the same space as their operands and make entire sentences comparable syntactically, and potentially semantically. The vectors generated had sufficient representational strength to generate parse trees at least comparable with contemporary symbolic parsers. | cs | 1 |
Title: Functional Inequalities for Brownian Motion on Riemannian Manifolds with Sticky-Reflecting Boundary Diffusion
Abstract: We prove geometric upper bounds for the Poincar\'e and Logarithmic Sobolev constants for Brownian motion on manifolds with sticky reflecting boundary diffusion i.e. extended Wentzell-type boundary condition under general curvature assumptions on the manifold and its boundary. The method is based on an interpolation involving energy interactions between the boundary and the interior of the manifold. As side results we obtain explicit geometric bounds on the first nontrivial Steklov eigenvalue, for the norm of the boundary trace operator on Sobolev functions, and on the boundary trace logarithmic Sobolev constant. The case of Brownian motion with pure sticky reflection is also treated. | math | 0 |
Title: More on homotopy continuation method and discounted zero-sum stochastic game with ARAT structure
Abstract: In this paper, we introduce a homotopy function to trace the trajectory by applying modified homotopy continuation method for finding the solution of two-person zero-sum discounted stochastic ARAT game. We show that the algorithm has the higher order of convergence. For the proposed algorithm, the homotopy path approaching the solution is smooth and bounded. Two numerical examples are illustrated to show the effectiveness of the proposed algorithm. | math | 1 |
Title: The cube axiom and resolutions in homotopy theory
Abstract: We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model structure. As an application, classical theorems in unstable homotopy theory are extended to this context. | math | 1 |
Title: Learned Image Downscaling for Upscaling using Content Adaptive Resampler
Abstract: Deep convolutional neural network based image super-resolution (SR) models have shown superior performance in recovering the underlying high resolution (HR) images from low resolution (LR) images obtained from the predefined downscaling methods. In this paper we propose a learned image downscaling method based on content adaptive resampler (CAR) with consideration on the upscaling process. The proposed resampler network generates content adaptive image resampling kernels that are applied to the original HR input to generate pixels on the downscaled image. Moreover, a differentiable upscaling (SR) module is employed to upscale the LR result into its underlying HR counterpart. By back-propagating the reconstruction error down to the original HR input across the entire framework to adjust model parameters, the proposed framework achieves a new state-of-the-art SR performance through upscaling guided image resamplers which adaptively preserve detailed information that is essential to the upscaling. Experimental results indicate that the quality of the generated LR image is comparable to that of the traditional interpolation based method, but the significant SR performance gain is achieved by deep SR models trained jointly with the CAR model. The code is publicly available on: URL https://github.com/sunwj/CAR. | cs | 1 |
Title: Spiking NeRF: Representing the Real-World Geometry by a Discontinuous Representation
Abstract: A crucial reason for the success of existing NeRF-based methods is to build a neural density field for the geometry representation via multiple perceptron layers (MLPs). MLPs are continuous functions, however, real geometry or density field is frequently discontinuous at the interface between the air and the surface. Such a contrary brings the problem of unfaithful geometry representation. To this end, this paper proposes spiking NeRF, which leverages spiking neurons and a hybrid Artificial Neural Network (ANN)-Spiking Neural Network (SNN) framework to build a discontinuous density field for faithful geometry representation. Specifically, we first demonstrate the reason why continuous density fields will bring inaccuracy. Then, we propose to use the spiking neurons to build a discontinuous density field. We conduct a comprehensive analysis for the problem of existing spiking neuron models and then provide the numerical relationship between the parameter of the spiking neuron and the theoretical accuracy of geometry. Based on this, we propose a bounded spiking neuron to build the discontinuous density field. Our method achieves SOTA performance. The source code and the supplementary material are available at https://github.com/liaozhanfeng/Spiking-NeRF. | cs | 0 |
Title: Pursuing the double affine Grassmannian III: Convolution with affine Zastava
Abstract: This is the third paper of a series (started by arXiv:0711.2083, arXiv:0908.3390) which describes a conjectural analog of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian and affine Zastava. | math | 1 |
Title: Free unitary groups are (almost) simple
Abstract: We show that the quotients of Wang and Van Daele's universal quantum groups by their centers are simple in the sense that they have no normal quantum subgroups, thus providing the first examples of simple compact quantum groups with non-commutative fusion rings. | math | 1 |
Title: Capacity Bounds and User Identification Costs in Rayleigh-Fading Many-Access Channel
Abstract: Many-access channel (MnAC) model allows the number of users in the system and the number of active users to scale as a function of the blocklength and as such is suited for dynamic communication systems with massive number of users such as the Internet of Things. Existing MnAC models assume a priori knowledge of channel gains which is impractical since acquiring Channel State Information (CSI) for massive number of users can overwhelm the available radio resources. This paper incorporates Rayleigh fading effects to the MnAC model and derives an upper bound on the symmetric message-length capacity of the Rayleigh-fading Gaussian MnAC. Furthermore, a lower bound on the minimum number of channel uses for discovering the active users is established. In addition, the performance of Noisy Combinatorial Orthogonal Matching Pursuit (N-COMP) based group testing (GT) is studied as a practical strategy for active device discovery. Simulations show that, for a given SNR, as the number of users increase, the required number of channel uses for N-COMP GT scales approximately the same way as the lower bound on minimum user identification cost. Moreover, in the low SNR regime, for sufficiently large population sizes, the number of channel uses required by N-COMP GT was observed to be within a factor of two of the lower bound when the expected number of active users scales sub-linearly with the total population size. | cs | 1 |
Title: Model theory and the Tannakian formalism
Abstract: We draw the connection between the model theoretic notions of internality and the binding group on one hand, and the Tannakian formalism on the other. More precisely, we deduce the fundamental results of the Tannakian formalism by associating to a Tannakian category a first order theory, and applying the results on internality there. We also formulate the notion of a differential tensor category, and a version of the Tannakian formalism for differential linear groups, and show how the same techniques can be used to deduce the analogous results in that context. | math | 1 |
Title: Sampling unknown large networks restricted by low sampling rates
Abstract: Graph sampling plays an important role in data mining for large networks. Specifically, larger networks often correspond to lower sampling rates. Under the situation, traditional traversal-based samplings for large networks usually have an excessive preference for densely-connected network core nodes. Aim at this issue, this paper proposes a sampling method for unknown networks at low sampling rates, called SLSR, which first adopts a random node sampling to evaluate a degree threshold, utilized to distinguish the core from periphery, and the average degree in unknown networks, and then runs a double-layer sampling strategy on the core and periphery. SLSR is simple that results in a high time efficiency, but experimental evaluation confirms that the proposed method can accurately preserve many critical structures of unknown large networks with low sampling rates and low variances. | cs | 0 |
Title: Global topology of the Hitchin system
Abstract: Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory. | math | 1 |
Title: Every closed surface of genus at least $17$ is Loewner
Abstract: In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus $g\geq 17$ satisfies Loewner's systolic ratio inequality. | math | 0 |
Title: Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras
Abstract: We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on $\mathbb{C}^n$ is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function $\mathbb{C}^n \to \mathbb{C}^n$ is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by "$C^1$ definable predicates" whose gradients are definable functions. As a consequence, we show that every element in the definable closure of $\mathrm{W}^*(x_1,\dots,x_n)$ can be expressed as a definable function of $(x_1,\dots,x_n)$. We give several classes of examples showing that the definable closure can be much larger than $\mathrm{W}^*(x_1,\dots,x_n)$ in general. | math | 0 |
Title: Consistent estimation of non-bandlimited spectral density from uniformly spaced samples
Abstract: In the matter of selection of sample time points for the estimation of the power spectral density of a continuous time stationary stochastic process, irregular sampling schemes such as Poisson sampling are often preferred over regular (uniform) sampling. A major reason for this preference is the well-known problem of inconsistency of estimators based on regular sampling, when the underlying power spectral density is not bandlimited. It is argued in this paper that, in consideration of a large sample property like consistency, it is natural to allow the sampling rate to go to infinity as the sample size goes to infinity. Through appropriate asymptotic calculations under this scenario, it is shown that the smoothed periodogram based on regularly spaced data is a consistent estimator of the spectral density, even when the latter is not band-limited. It transpires that, under similar assumptions, the estimators based on uniformly sampled and Poisson-sampled data have about the same rate of convergence. Apart from providing this reassuring message, the paper also gives a guideline for practitioners regarding appropriate choice of the sampling rate. Theoretical calculations for large samples and Monte-Carlo simulations for small samples indicate that the smoothed periodogram based on uniformly sampled data have less variance and more bias than its counterpart based on Poisson sampled data. | math | 1 |
Title: A Universal Lipschitz Extension Property of Gromov Hyperbolic Spaces
Abstract: A metric space has the universal Lipschitz extension property if for each subspace S embedded quasi-isometrically into an arbitrary metric space M there exists a continuous linear extension of Banach-valued Lipschitz functions on S to those on all of M. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition. | math | 1 |
Title: A Generative AI Assistant to Accelerate Cloud Migration
Abstract: We present a tool that leverages generative AI to accelerate the migration of on-premises applications to the cloud. The Cloud Migration LLM accepts input from the user specifying the parameters of their migration, and outputs a migration strategy with an architecture diagram. A user study suggests that the migration LLM can assist inexperienced users in finding the right cloud migration profile, while avoiding complexities of a manual approach. | cs | 0 |
Title: Extreme statistics of non-intersecting Brownian paths
Abstract: We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [MFQR13] for the joint distribution of $\mathcal{M}={\rm max}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$ and $\mathcal{T}={\rm argmax}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$, where $\mathcal{A}_2$ is the Airy$_2$ process, and we use them to show that in the three cases the joint distribution converges, as $N\to\infty$, to the joint distribution of $\mathcal{M}$ and $\mathcal{T}$. In the case of non-intersecting Brownian bridges on the line, we also establish small deviation inequalities for the argmax which match the tail behavior of $\mathcal{T}$. Our proofs are based on the method introduced in [CQR13,BCR15] for obtaining formulas for the probability that the top line of these line ensembles stays below a given curve, which are given in terms of the Fredholm determinant of certain "path-integral" kernels. | math | 1 |
Title: Representing maps for semibounded forms and their Lebesgue type decompositions
Abstract: For a semibounded sesquilinear form ${\mathfrak t}$ in a Hilbert space ${\mathfrak H}$ there exists a representing map $Q$ from ${\mathfrak H}$ to another Hilbert space ${\mathfrak K}$, such that ${\mathfrak t}[\varphi, \psi]-c(\varphi, \psi)=(Q\varphi,Q\psi)$, $\varphi,\psi \in {\rm dom\,}{\mathfrak t}$, with $c \in {\mathbb R}$ a lower bound of ${\mathfrak t}$. Representing maps offer a simplifying tool to study general semibounded forms. By means of representing maps closedness, closability, and singularity of ${\mathfrak t}$ are immediately translated into the corresponding properties of the operator $Q$, and vice versa. Also properties of sum decompositions ${\mathfrak t}={\mathfrak t}_1+{\mathfrak t}_2$ of a nonnegative form ${\mathfrak t}$ with two other nonnegative forms ${\mathfrak t}_1$ and ${\mathfrak t}_2$ in ${\mathfrak H}$ can be analyzed by means of associated nonnegative contractions $K\in {\mathbf B}({\mathfrak K})$. This helps, for instance, to establish an explicit operator theoretic characterization for the summands ${\mathfrak t}_1$ and ${\mathfrak t}_2$ to be, or not to be, mutually singular. Such sum decompositions are used to study characteristic properties of the so-called Lebesgue type decompositions of semibounded forms ${\mathfrak t}$, where ${\mathfrak t}_1$ is closable and ${\mathfrak t}_2$ singular; in particular, this includes the Lebesgue decomposition of a semibounded form due to B. Simon. Furthermore, for a semibounded form ${\mathfrak t}$ with its representing map $Q$ it will be shown that the corresponding semibounded selfadjoint relation $Q^*Q^{**} +c$ is uniquely determined by a limit version of the classical representation theorem for the form ${\mathfrak t}$, being studied by W. Arendt and T. ter Elst in a sectorial context. Via representing maps a full treatment is given of the convergence of monotone sequences of semibounded forms. | math | 0 |
Title: L3Cube-IndicNews: News-based Short Text and Long Document Classification Datasets in Indic Languages
Abstract: In this work, we introduce L3Cube-IndicNews, a multilingual text classification corpus aimed at curating a high-quality dataset for Indian regional languages, with a specific focus on news headlines and articles. We have centered our work on 10 prominent Indic languages, including Hindi, Bengali, Marathi, Telugu, Tamil, Gujarati, Kannada, Odia, Malayalam, and Punjabi. Each of these news datasets comprises 10 or more classes of news articles. L3Cube-IndicNews offers 3 distinct datasets tailored to handle different document lengths that are classified as: Short Headlines Classification (SHC) dataset containing the news headline and news category, Long Document Classification (LDC) dataset containing the whole news article and the news category, and Long Paragraph Classification (LPC) containing sub-articles of the news and the news category. We maintain consistent labeling across all 3 datasets for in-depth length-based analysis. We evaluate each of these Indic language datasets using 4 different models including monolingual BERT, multilingual Indic Sentence BERT (IndicSBERT), and IndicBERT. This research contributes significantly to expanding the pool of available text classification datasets and also makes it possible to develop topic classification models for Indian regional languages. This also serves as an excellent resource for cross-lingual analysis owing to the high overlap of labels among languages. The datasets and models are shared publicly at https://github.com/l3cube-pune/indic-nlp | cs | 0 |
Title: Induction of quantum group representations
Abstract: Induced representations for quantum groups are defined starting from coisotropic quantum subgroups and their main properties are proved. When the coisotropic quantum subgroup has a suitably defined section such representations can be realized on associated quantum bundles on general embeddable quantum homogeneous spaces. | math | 1 |
Title: Edge detection with trigonometric polynomial shearlets
Abstract: In this paper we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vall\'{e}e Poussin type wavelets are able to detect singularities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of corresponding inner products. In the proof we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results. | math | 1 |
Title: On the Hankel Transform of Bessel Functions on Complex Numbers and Explicit Spectral Formulae over the Gaussian Field
Abstract: In this paper, on the complex field $\mathbb{C}$, we prove two integral formulae for the Hankel-Mellin transform and the double Fourier-Mellin transform of Bessel functions, both resulting the hypergeometric function. As two applications, we use the former integral formula to make explicit the spectral formula of Bruggeman and Motohashi for the fourth moment of Dedekind zeta function over the Gaussian number field $\mathbb{Q}(i)$ and to establish a spectral formula for the Hecke-eigenvalue twisted second moment of central $L$-values for the Picard group $\mathrm{PGL}_2 (\mathbb{Z}[i])$. Moreover, we develop the theory of distributional Hankel transform on $\mathbb{C} \smallsetminus \{0\}$. | math | 0 |
Title: Global uniqueness of small representations
Abstract: We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper. | math | 1 |
Title: Linearly shellable complexes
Abstract: We introduce the class of linearly shellable pure simplicial complexes. The characterizing property is the existence of a labeling of their vertices such that all linear extensions of the Bruhat order on the set of facets are shelling orders. Coxeter complexes of weak intervals in Coxeter groups of finite rank are proved to be linearly shellable. We also introduce the notion of linear strong shellability. | math | 0 |
Title: Parameterized Verification of Disjunctive Timed Networks
Abstract: We introduce new techniques for the parameterized verification of disjunctive timed networks (DTNs), i.e., networks of timed automata (TAs) that communicate via location guards that enable a transition only if there is another process in a given location. This computational model has been considered in the literature before, example applications are gossiping clock synchronization protocols or planning problems. We address the minimum-time reachability problem (Minreach) in DTNs, and show how to efficiently solve it based on a novel zone graph algorithm. We further show that solving Minreach allows us to construct a summary TA capturing exactly the possible behaviors of a single TA within a DTN of arbitrary size. The combination of these two results enables the parameterized verification of DTNs, while avoiding the construction of an exponential-size cutoff system required by existing results. Additionally, we develop sufficient conditions for solving Minreach and parameterized verification problems even in certain cases where locations that appear in location guards can have clock invariants, a case that has usually been excluded in the literature. Our techniques are also implemented, and experiments show their practicality. | cs | 0 |
Title: Algorithms Transcending the SAT-Symmetry Interface
Abstract: Dedicated treatment of symmetries in satisfiability problems (SAT) is indispensable for solving various classes of instances arising in practice. However, the exploitation of symmetries usually takes a black box approach. Typically, off-the-shelf external, general-purpose symmetry detection tools are invoked to compute symmetry groups of a formula. The groups thus generated are a set of permutations passed to a separate tool to perform further analyzes to understand the structure of the groups. The result of this second computation is in turn used for tasks such as static symmetry breaking or dynamic pruning of the search space. Within this pipeline of tools, the detection and analysis of symmetries typically incurs the majority of the time overhead for symmetry exploitation. In this paper we advocate for a more holistic view of what we call the SAT-symmetry interface. We formulate a computational setting, centered around a new concept of joint graph/group pairs, to analyze and improve the detection and analysis of symmetries. Using our methods, no information is lost performing computational tasks lying on the SAT-symmetry interface. Having access to the entire input allows for simpler, yet efficient algorithms. Specifically, we devise algorithms and heuristics for computing finest direct disjoint decompositions, finding equivalent orbits, and finding natural symmetric group actions. Our algorithms run in what we call instance-quasi-linear time, i.e., almost linear time in terms of the input size of the original formula and the description length of the symmetry group returned by symmetry detection tools. Our algorithms improve over both heuristics used in state-of-the-art symmetry exploitation tools, as well as theoretical general-purpose algorithms. | cs | 0 |
Title: Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems
Abstract: We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blow-up formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizio\l~ on log $K$-theory. Using the resulting \emph{saturated descent}, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for $A_{\inf}$-cohomology. | math | 0 |
Title: Bernstein components via Bernstein center
Abstract: Let G be a reductive p-adic group. Let $\Phi$ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that $\Phi$ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modiification of the arguments of J.F.Dat who proved our result in one direction. | math | 1 |
Title: Playing Atari with Six Neurons
Abstract: Deep reinforcement learning, applied to vision-based problems like Atari games, maps pixels directly to actions; internally, the deep neural network bears the responsibility of both extracting useful information and making decisions based on it. By separating the image processing from decision-making, one could better understand the complexity of each task, as well as potentially find smaller policy representations that are easier for humans to understand and may generalize better. To this end, we propose a new method for learning policies and compact state representations separately but simultaneously for policy approximation in reinforcement learning. State representations are generated by an encoder based on two novel algorithms: Increasing Dictionary Vector Quantization makes the encoder capable of growing its dictionary size over time, to address new observations as they appear in an open-ended online-learning context; Direct Residuals Sparse Coding encodes observations by disregarding reconstruction error minimization, and aiming instead for highest information inclusion. The encoder autonomously selects observations online to train on, in order to maximize code sparsity. As the dictionary size increases, the encoder produces increasingly larger inputs for the neural network: this is addressed by a variation of the Exponential Natural Evolution Strategies algorithm which adapts its probability distribution dimensionality along the run. We test our system on a selection of Atari games using tiny neural networks of only 6 to 18 neurons (depending on the game's controls). These are still capable of achieving results comparable---and occasionally superior---to state-of-the-art techniques which use two orders of magnitude more neurons. | cs | 1 |
Title: Synthetic projective lines, geometric closure and AB-sets
Abstract: In this note, we introduce a new approach to abstract ``synthetic'' projective lines. We discuss various aspects of our approach, and compare these aspects with the classical one. A number of intriguing questions arise. Amongst these aspects, we discuss geometric closures, and introduce automorphism-blocking sets. We also construct a number of (counter) examples in infinite cases. | math | 1 |
Title: Language Models Are Greedy Reasoners: A Systematic Formal Analysis of Chain-of-Thought
Abstract: Large language models (LLMs) have shown remarkable reasoning capabilities given chain-of-thought prompts (examples with intermediate reasoning steps). Existing benchmarks measure reasoning ability indirectly, by evaluating accuracy on downstream tasks such as mathematical reasoning. However, it is unclear how these models obtain the answers and whether they rely on simple heuristics rather than the generated chain-of-thought. To enable systematic exploration of the reasoning ability of LLMs, we present a new synthetic question-answering dataset called PrOntoQA, where each example is generated from a synthetic world model represented in first-order logic. This allows us to parse the generated chain-of-thought into symbolic proofs for formal analysis. Our analysis on InstructGPT and GPT-3 shows that LLMs are quite capable of making correct individual deduction steps, and so are generally capable of reasoning, even in fictional contexts. However, they have difficulty with proof planning: When multiple valid deduction steps are available, they are not able to systematically explore the different options. | cs | 1 |
Title: An Edge-Cloud Collaboration Framework for Generative AI Service Provision with Synergetic Big Cloud Model and Small Edge Models
Abstract: Generative artificial intelligence (GenAI) offers various services to users through content creation, which is believed to be one of the most important components in future networks. However, training and deploying big artificial intelligence models (BAIMs) introduces substantial computational and communication overhead.This poses a critical challenge to centralized approaches, due to the need of high-performance computing infrastructure and the reliability, secrecy and timeliness issues in long-distance access of cloud services. Therefore, there is an urging need to decentralize the services, partly moving them from the cloud to the edge and establishing native GenAI services to enable private, timely, and personalized experiences. In this paper, we propose a brand-new bottom-up BAIM architecture with synergetic big cloud model and small edge models, and design a distributed training framework and a task-oriented deployment scheme for efficient provision of native GenAI services. The proposed framework can facilitate collaborative intelligence, enhance adaptability, gather edge knowledge and alleviate edge-cloud burden. The effectiveness of the proposed framework is demonstrated through an image generation use case. Finally, we outline fundamental research directions to fully exploit the collaborative potential of edge and cloud for native GenAI and BAIM applications. | cs | 0 |
Title: Which Quantum Circuit Mutants Shall Be Used? An Empirical Evaluation of Quantum Circuit Mutations
Abstract: As a new research area, quantum software testing lacks systematic testing benchmarks to assess testing techniques' effectiveness. Recently, some open-source benchmarks and mutation analysis tools have emerged. However, there is insufficient evidence on how various quantum circuit characteristics (e.g., circuit depth, number of quantum gates), algorithms (e.g., Quantum Approximate Optimization Algorithm), and mutation characteristics (e.g., mutation operators) affect the most mutant detection in quantum circuits. Studying such relations is important to systematically design faulty benchmarks with varied attributes (e.g., the difficulty in detecting a seeded fault) to facilitate assessing the cost-effectiveness of quantum software testing techniques efficiently. To this end, we present a large-scale empirical evaluation with more than 700K faulty benchmarks (quantum circuits) generated by mutating 382 real-world quantum circuits. Based on the results, we provide valuable insights for researchers to define systematic quantum mutation analysis techniques. We also provide a tool to recommend mutants to users based on chosen characteristics (e.g., a quantum algorithm type) and the required difficulty of killing mutants. Finally, we also provide faulty benchmarks that can already be used to assess the cost-effectiveness of quantum software testing techniques. | cs | 0 |
Title: Mean Oriented Riesz Features for Micro Expression Classification
Abstract: Micro-expressions are brief and subtle facial expressions that go on and off the face in a fraction of a second. This kind of facial expressions usually occurs in high stake situations and is considered to reflect a human's real intent. There has been some interest in micro-expression analysis, however, a great majority of the methods are based on classically established computer vision methods such as local binary patterns, histogram of gradients and optical flow. A novel methodology for micro-expression recognition using the Riesz pyramid, a multi-scale steerable Hilbert transform is presented. In fact, an image sequence is transformed with this tool, then the image phase variations are extracted and filtered as proxies for motion. Furthermore, the dominant orientation constancy from the Riesz transform is exploited to average the micro-expression sequence into an image pair. Based on that, the Mean Oriented Riesz Feature description is introduced. Finally the performance of our methods are tested in two spontaneous micro-expressions databases and compared to state-of-the-art methods. | cs | 1 |
Title: Perfect numbers and Fibonacci primes
Abstract: In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that $\sum_{d|n,d<n}d^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$ and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers. | math | 1 |
Title: An upwind method for genuine weakly hyperbolic systems
Abstract: In this article, we attempted to develop an upwind scheme based on Flux Difference Splitting using Jordan canonical forms to simulate genuine weakly hyperbolic systems. Theory of Jordan Canonical Forms is being used to complete defective set of linear independent eigenvectors. Proposed FDS-J scheme is capable of recognizing various shocks accurately. | math | 1 |
Title: Forced translational symmetry-breaking for abstract evolution equations: the organizing center for blocking of travelling waves
Abstract: We consider two parameter families of differential equations on a Banach space X, where the parameters c and $\epsilon$ are such that: (1) when $\epsilon=0$, the differential equations are symmetric under the action of the group of one-dimensional translations SE(1) acting on X, whereas when $\epsilon\neq 0$, this translation symmetry is broken, (2) when $\epsilon=0$, the symmetric differential equations admit a smooth family of relative equilibria (travelling waves) parametrized by the drift speed c, with $c=0$ corresponding to steady-states. Under certain hypotheses on the differential equations and on the Banach space X, we use the center manifold theorem of Sandstede, Scheel and Wulff to study the effects of the symmetry-breaking perturbation on the above family of relative equilibria. In particular, we show that the phenomenon commonly referred to as propagation failure, or wave blocking occurs in a cone in the $(c,\epsilon)$ parameter space which emanates from the point $(c,\epsilon)=(0,0)$. We also discuss how our methods can be adapted to perturbations of parameter-independent differential equations (such as the Fisher-KPP) which admit families of relative equilibria parametrized by drift speed. | math | 1 |
Title: Cohomology of fixed point sets of anti-symplectic involutions in the Hilbert scheme of points on a surface
Abstract: Let $S$ be a smooth, quasi-projective complex surface with complex symplectic form $\omega \in H^0(S, K_S)$. This determines a symplectic form $\omega_n$ on the Hilbert scheme of points $S^{[n]}$ for $n \geq 1$. Let $\tau$ be an anti-symplectic involution of $(S,\omega)$: an order two automorphism of $S$ such that $ \tau^*\omega=-\omega$. Then $\tau$ induces an anti-symplectic involution on $(S^{[n]},\omega_n)$ and the fixed point set $(S^{[n]})^\tau$ is a smooth Lagrangian subvariety of $S^{[n]}$. In this paper, we calculate the mixed Hodge structure of $H^*( (S^{[n]})^\tau; \mathbb{Q})$ in terms of the mixed Hodge structures of $H^*( S^\tau;\mathbb{Q})$ and of $H^*( S / \tau; \mathbb{Q})$. We also classify the connected components of $(S^{[n]})^\tau$ and determine their mixed Hodge structures. Our results apply more generally whenever $S$ is a smooth quasi-projective surface, and $\tau$ is an involution of $S$ for which $S^\tau$ is a curve. | math | 0 |
Title: Recognition of Unit Segment and Polyline Graphs is $\exists\mathbb{R}$-Complete
Abstract: Given a set of objects O in the plane, the corresponding intersection graph is defined as follows. A vertex is created for each object and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly k bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as the intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $\exists\mathbb{R}$-complete, leaving unit segments and polylines as few remaining natural cases. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete. | cs | 0 |
Title: Paraconsistent Existential Graphs Gamma Peirce System
Abstract: In this paper, the paraconsistent propositional logic LG is presented, along with its semantic characterization. It is shown that LG's set of theorems corresponds to the set of valid existential graphs, GET, which turns out to be an extension of Peirce's Gamma system, without becoming Zeman's gamma-4 system. All evidence is presented in a complete, rigorous, and detailed manner. This result is generalized by constructing the paraconsistent system of existential graphs GET4, and its semantic-deductive characterization. Finally, Zeman's Gamma-4, Gamma-4.2, and Gamma-5 existential graph systems are proven to be paraconsistent. | math | 0 |
Title: Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings
Abstract: This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the $(k,q)$-colourings of a graph $G$ can be obtained by successively recolouring a single vertex provided $k/q\geq 2col(G)$ along the lines of Cereceda, van den Heuvel and Johnson's result for $k$-colourings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colourings coincide. As a corollary, we obtain an Albertson-type extension theorem for $(k,q)$-precolourings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs $G$ for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role. | math | 1 |
Title: Parabolic bifurcation loci in the spaces of rational functions
Abstract: We give a geometric description of the parabolic bifurcation locus in the space $\operatorname{Rat}_d$ of all rational functions on $\mathbb{P}^1$ of degree $d>1$, generalizing the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions. | math | 0 |
Title: Longest and shortest cycles in random planar graphs
Abstract: Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in $P(n,m)$ in the critical range when $m=n/2+o(n)$. In addition, we describe the block structure of $P(n,m)$ in the weakly supercritical regime when $n^{2/3}\ll m-n/2\ll n$. | math | 1 |
Title: Rewrite Caption Semantics: Bridging Semantic Gaps for Language-Supervised Semantic Segmentation
Abstract: Vision-Language Pre-training has demonstrated its remarkable zero-shot recognition ability and potential to learn generalizable visual representations from language supervision. Taking a step ahead, language-supervised semantic segmentation enables spatial localization of textual inputs by learning pixel grouping solely from image-text pairs. Nevertheless, the state-of-the-art suffers from clear semantic gaps between visual and textual modality: plenty of visual concepts appeared in images are missing in their paired captions. Such semantic misalignment circulates in pre-training, leading to inferior zero-shot performance in dense predictions due to insufficient visual concepts captured in textual representations. To close such semantic gap, we propose Concept Curation (CoCu), a pipeline that leverages CLIP to compensate for the missing semantics. For each image-text pair, we establish a concept archive that maintains potential visually-matched concepts with our proposed vision-driven expansion and text-to-vision-guided ranking. Relevant concepts can thus be identified via cluster-guided sampling and fed into pre-training, thereby bridging the gap between visual and textual semantics. Extensive experiments over a broad suite of 8 segmentation benchmarks show that CoCu achieves superb zero-shot transfer performance and greatly boosts language-supervised segmentation baseline by a large margin, suggesting the value of bridging semantic gap in pre-training data. | cs | 0 |
Title: The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets
Abstract: In this paper, we consider the planar self-affine measures $\mu_{M,D}$ generated by an expanding matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $ D=\left\{ {\left( {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} \alpha_1\\ \alpha_2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} \beta_1\\ \beta_2 \end{array}} \right)} \right\} $ with $\alpha_1\beta_2-\alpha_2\beta_1\neq0$. We show that if $\det(M)\notin 3\mathbb{Z}$, then the mutually orthogonal exponential functions in $L^2(\mu_{M,D})$ is finite, and the exact maximal cardinality is given. | math | 1 |
Title: Gromov-Witten Invariants of Bielliptic Surfaces
Abstract: Bielliptic surfaces appear as quotient of a product of two elliptic curves and were classified by Bagnera-Franchis. We give a concrete way of computing their GW-invariants with point insertions using a floor diagram algorithm. Using the latter, we are able to prove the quasi-modularity of their generating series by relating them to generating series of graphs for which we also prove quasi-modularity results. We propose a refinement of these invariants by inserting a {\lambda}-class in the considered GW-invariants. | math | 0 |
Title: A note on the equidistribution of $3$-colour partitions
Abstract: In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(\zeta ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- \zeta {\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $\zeta$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand. | math | 0 |
Title: Continuum dynamics of the intention field under weakly cohesive social interactions
Abstract: We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. Firstly, we derive a Fokker-Planck type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Secondly, we study conditions under which the Fokker-Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence. | math | 1 |
Title: Global well-posedness for the Euler alignment system with mildly singular interactions
Abstract: We consider the Euler alignment system with mildly singular interaction kernels. When the local repulsion term is of the fractional type, global in time existence of smooth solutions was proved in\cite{do2018global,shvydkoy2017eulerian1,shvydkoy2017eulerian2,shvydkoy2017eulerian3}. Here, we consider a class of less singular interaction kernels and establish the global regularity of solutions as long as the interaction kernels are not integrable. The proof relies on modulus of continuity estimates for a class of parabolic integro-differential equations with a drift and mildly singular kernels. | math | 1 |
Title: Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II)
Abstract: We consider the nonlinear system $c_1p_1^d +c_2p_2^d + \dots + c_s p_s^d = 0$ with $c_1, c_2,\dots, c_s\in\mathbb Z$ being nonzero and satisfying $c_1 +c_2 + \dots + c_s = 0$. We show that for $s\ge 2\lfloor \frac{d^2}2\rfloor+1$ and $c\in\left(1, 1+c(d,s)\right)$, if the system has only $K$-trivial solutions in subset $\mathcal{A}$ of Piatetski-Shapiro primes up to $x$ and corresponding to $c$, then $|\mathcal{A}| \ll \frac{x^{\frac1c}}{\log x} $$\left(\log \log \log \log x\right)^{\frac{2-s}{dc}+\varepsilon}$. | math | 0 |
Title: Theoretical guarantees on the best-of-n alignment policy
Abstract: A simple and effective method for the alignment of generative models is the best-of-$n$ policy, where $n$ samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-$n$ policy and the base policy is equal to $\log (n) - (n-1)/n.$ We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples. | cs | 0 |
Title: On the heavy-tail behavior of the distributionally robust newsvendor
Abstract: Since the seminal work of Scarf (1958) [A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production, pages 201-209] on the newsvendor problem with ambiguity in the demand distribution, there has been a growing interest in the study of the distributionally robust newsvendor problem. The model is criticized at times for being overly conservative since the worst-case distribution is discrete with a few support points. However, it is the order quantity prescribed from the model that is of practical relevance. A simple calculation shows that the optimal order quantity in Scarf's model with known first and second moment is also optimal for a censored student-t distribution with parameter 2. In this paper, we generalize this "heavy-tail optimality" property of the distributionally robust newsvendor to an ambiguity set where information on the first and the $\alpha$th moment is known, for any real number $\alpha > 1$. We show that the optimal order quantity for the distributionally robust newsvendor problem is also optimal for a regularly varying distribution with roughly a power law tail with tail index $\alpha$. We illustrate the usefulness of the model in the high service level regime with numerical experiments, by showing that when a standard distribution such as the exponential or lognormal distribution is contaminated with a heavy-tailed (regularly varying) distribution, the distributionally robust optimal order quantity outperforms the optimal order quantity of the original distribution, even with a small amount of contamination. | math | 1 |
Title: Toroidal orbifolds, destackification, and Kummer blowings up
Abstract: We show that any toroidal DM stack $X$ with finite diagonalizable inertia possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X\to X_{tcs}$ is logarithmically smooth. Further, we use torification results of [AT17] to construct a destackification functor, a variant of the main result of Bergh [Ber17], on the category of such toroidal stacks $X$. Namely, we associate to $X$ a sequence of blowings up of toroidal stacks $\widetilde{\mathcal{F}}_X\:Y\longrightarrow X$ such that $Y_{tc}$ coincides with the usual coarse moduli space $Y_{cs}$. In particular, this provides a toroidal resolution of the algebraic space $X_{cs}$. Both $X_{tcs}$ and $\widetilde{\mathcal{F}}_X$ are functorial with respect to strict inertia preserving morphisms $X'\to X$. Finally, we use coarsening morphisms to introduce a class of non-representable birational modifications of toroidal stacks called Kummer blowings up. These modifications, as well as our version of destackification, are used in our work on functorial toroidal resolution of singularities. | math | 1 |
Title: Radio Network Lower Bounds Made Easy
Abstract: Theoreticians have studied distributed algorithms in the radio network model for close to three decades. A significant fraction of this work focuses on lower bounds for basic communication problems such as wake-up (symmetry breaking among an unknown set of nodes) and broadcast (message dissemination through an unknown network topology). In this paper, we introduce a new technique for proving this type of bound, based on reduction from a probabilistic hitting game, that simplifies and strengthens much of this existing work. In more detail, in this single paper we prove new expected time and high probability lower bounds for wake-up and global broadcast in single and multichannel versions of the radio network model both with and without collision detection. In doing so, we are able to reproduce results that previously spanned a half-dozen papers published over a period of twenty-five years. In addition to simplifying these existing results, our technique, in many places, also improves the state of the art: of the eight bounds we prove, four strictly strengthen the best known previous result (in terms of time complexity and/or generality of the algorithm class for which it holds), and three provide the first known non-trivial bound for the case in question. The fact that the same technique can easily generate this diverse collection of lower bounds indicates a surprising unity underlying communication tasks in the radio network model---revealing that deep down, below the specifics of the problem definition and model assumptions, communication in this setting reduces to finding efficient strategies for a simple game. | cs | 1 |
Title: Rigidity And Unirational Groups
Abstract: We prove a rigidity theorem for morphisms from products of open subschemes of the projective line into solvable groups not containing a copy of $\Ga$ (for example, wound unipotent groups). As a consequence, we deduce several structural results about unirational group schemes, including that unirationality for group schemes descends through separable extensions. We also apply the main result to prove that permawound unipotent groups are unirational and -- when wound -- commutative. | math | 0 |
Title: A parametricity-based formalization of semi-simplicial and semi-cubical sets
Abstract: Semi-simplicial and semi-cubical sets are commonly defined as presheaves over respectively, the semi-simplex or semi-cube category. Homotopy Type Theory then popularized an alternative definition, where the set of n-simplices or n-cubes are instead regrouped into the families of the fibers over their faces, leading to a characterization we call indexed. Moreover, it is known that semi-simplicial and semi-cubical sets are related to iterated Reynolds parametricity, respectively in its unary and binary variants. We exploit this correspondence to develop an original uniform indexed definition of both augmented semi-simplicial and semi-cubical sets, and fully formalize it in Coq. | cs | 0 |
Title: Dynamically Masked Discriminator for Generative Adversarial Networks
Abstract: Training Generative Adversarial Networks (GANs) remains a challenging problem. The discriminator trains the generator by learning the distribution of real/generated data. However, the distribution of generated data changes throughout the training process, which is difficult for the discriminator to learn. In this paper, we propose a novel method for GANs from the viewpoint of online continual learning. We observe that the discriminator model, trained on historically generated data, often slows down its adaptation to the changes in the new arrival generated data, which accordingly decreases the quality of generated results. By treating the generated data in training as a stream, we propose to detect whether the discriminator slows down the learning of new knowledge in generated data. Therefore, we can explicitly enforce the discriminator to learn new knowledge fast. Particularly, we propose a new discriminator, which automatically detects its retardation and then dynamically masks its features, such that the discriminator can adaptively learn the temporally-vary distribution of generated data. Experimental results show our method outperforms the state-of-the-art approaches. | cs | 0 |
Title: Communication games, sequential equilibrium, and mediators
Abstract: We consider $k$-resilient sequential equilibria, strategy profiles where no player in a coalition of at most $k$ players believes that it can increase its utility by deviating, regardless of its local state. We prove that all $k$-resilient sequential equilibria that can be implemented with a trusted mediator can also be implemented without the mediator in a synchronous system of $n$ players if $n >3k$. In asynchronous systems, where there is no global notion of time and messages may take arbitrarily long to get to their recipient, we prove that a $k$-resilient sequential equilibrium with a mediator can be implemented without the mediator if $n > 4k$. These results match the lower bounds given by Abraham, Dolev, and Halpern (2008) and Geffner and Halpern (2023) for implementing a Nash equilibrium without a mediator (which are easily seen to apply to implementing a sequential equilibrium) and improve the results of Gerardi, who showed that, in the case that $k=1$, a sequential equilibrium can be implemented in synchronous systems if $n \ge 5$. | cs | 0 |
Title: Training-free Content Injection using h-space in Diffusion Models
Abstract: Diffusion models (DMs) synthesize high-quality images in various domains. However, controlling their generative process is still hazy because the intermediate variables in the process are not rigorously studied. Recently, the bottleneck feature of the U-Net, namely $h$-space, is found to convey the semantics of the resulting image. It enables StyleCLIP-like latent editing within DMs. In this paper, we explore further usage of $h$-space beyond attribute editing, and introduce a method to inject the content of one image into another image by combining their features in the generative processes. Briefly, given the original generative process of the other image, 1) we gradually blend the bottleneck feature of the content with proper normalization, and 2) we calibrate the skip connections to match the injected content. Unlike custom-diffusion approaches, our method does not require time-consuming optimization or fine-tuning. Instead, our method manipulates intermediate features within a feed-forward generative process. Furthermore, our method does not require supervision from external networks. The code is available at https://curryjung.github.io/InjectFusion/ | cs | 0 |
Title: Multi-task learning for Joint Language Understanding and Dialogue State Tracking
Abstract: This paper presents a novel approach for multi-task learning of language understanding (LU) and dialogue state tracking (DST) in task-oriented dialogue systems. Multi-task training enables the sharing of the neural network layers responsible for encoding the user utterance for both LU and DST and improves performance while reducing the number of network parameters. In our proposed framework, DST operates on a set of candidate values for each slot that has been mentioned so far. These candidate sets are generated using LU slot annotations for the current user utterance, dialogue acts corresponding to the preceding system utterance and the dialogue state estimated for the previous turn, enabling DST to handle slots with a large or unbounded set of possible values and deal with slot values not seen during training. Furthermore, to bridge the gap between training and inference, we investigate the use of scheduled sampling on LU output for the current user utterance as well as the DST output for the preceding turn. | cs | 1 |
Title: Stable matchings with correlated Preferences
Abstract: The stable matching problem has been the subject of intense theoretical and empirical study since the seminal 1962 paper by Gale and Shapley. The number of stable matchings for different systems of preferences has been studied in many contexts, going back to Donald Knuth in the 1970s. In this paper, we consider a family of distributions defined by the Mallows permutations and show that with high probability the number of stable matchings for these preferences is exponential in the number of people. | math | 0 |
Title: Noncentral moderate deviations for time-changed Lévy processes with inverse of stable subordinators
Abstract: In this paper we present some extensions of recent noncentral moderate deviation results in the literature. In the first part we generalize the results in \cite{BeghinMacciSPL2022} by considering a general L\'evy process $\{S(t):t\geq 0\}$ instead of a compound Poisson process. In the second part we assume that $\{S(t):t\geq 0\}$ has bounded variation and it is not a subordinator; thus, in some sense, we have the difference of two independent non-null subordinators. In this way we generalize the results in \cite{LeeMacci} for Skellam processes. | math | 0 |
Title: A Complete Landscape for the Price of Envy-Freeness
Abstract: We study the efficiency of fair allocations using the well-studied price of fairness concept, which quantitatively measures the worst-case efficiency loss when imposing fairness constraints. Previous works provided partial results on the price of fairness with well-known fairness notions such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). In this paper, we give a complete characterization for the price of envy-freeness in various settings. In particular, we first consider the two-agent case under the indivisible-goods setting and present tight ratios for the price of EF1 (for scaled utility) and EFX (for unscaled utility), which resolve questions left open in the literature. Next, we consider the mixed goods setting which concerns a mixture of both divisible and indivisible goods. We focus on envy-freeness for mixed goods (EFM), which generalizes both envy-freeness and EF1, as well as its strengthening called envy-freeness up to any good for mixed goods (EFXM), which generalizes envy-freeness and EFX. To this end, we settle the price of EFM and EFXM by providing a complete picture of tight bounds for two agents and asymptotically tight bounds for $n$ agents, for both scaled and unscaled utilities. | cs | 0 |
Title: Energy Identity for Stationary Harmonic Maps
Abstract: In this paper we consider sequences $u_j:B_2\subseteq M\to N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j|^2\leq \Lambda$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+\nu$, where $\nu = e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(\nu)$ one can produce a finite number of bubble maps $b_j:S^2\to N$ by blowing up the sequence $u_j$ near $x$. We prove the energy identity in this paper. Namely, we have at a.e. $x\in S$ that $e(x)=\sum_j E[b_j]$ for a complete set of such bubbles. That is, the energy density of the defect measure $\nu$ is precisely the sum of the energies of the bubbling maps. | math | 0 |
Title: Group actions on stacks and applications to equivariant string topology for stacks
Abstract: This paper is a continuations of the project initiated in the book string topology for stacks. We construct string operations on the SO(2)-equivariant homology of the (free) loop space $L(X)$ of an oriented differentiable stack $X$ and show that $H^{SO(2)}_{*+dim(X) -2}(L(X))$ is a graded Lie algebra. In the particular case where $X$ is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on. | math | 1 |
Title: 3D printed architected lattice structures by material jetting
Abstract: High-precision 3D printing technology opens to almost endless opportunities to design complex shapes present in tailored architected materials. The scope of this work is to review the latest studies regarding 3D printed lattice structures that involve the use of photopolymers fabricated by Material Jetting (MJ), with a focus on the widely used Polyjet and MultiJet techniques. The main aspects governing this printing process are introduced to determine their influence during the fabrication of 3D printed lattices. Performed experimental studies, considered assumptions, and constitutive models for the respective numerical simulations are analyzed. Furthermore, an overview of the latest extensively studied 3D printed architected lattice materials is exposed by emphasizing their achieved mechanical performances through the use of Ashby plots. Then, we highlight the advantages, limitations, and challenges of the material jetting technology to manufacture tunable architected materials for innovative devices, oriented to several engineering applications. Finally, possible approaches for future works and gaps to be covered by further research are indicated, including cost and environmental-related issues. | cs | 1 |
Title: Bases for free Lie superalgebras
Abstract: We describe a basis for free Lie superalgebras which uses the theory of basic commutators. The only description for bases for free Lie superalgebras that I have found in the literature is in the book "Infinite dimensional Lie superalgebras" by Bahturin et al. Their bases make use of the theory of Shirshov bases in free Lie algebras, and I believe that there is a case for writing up an alternative approach using basic commutators. | math | 0 |
Title: Symplectic reduction and lagrangian submanifolds in Gr(1, n)
Abstract: We study lagrangian submanifolds of algebraic variety Gr(1, n) equipped with the Kahler form given by the Plucker embedding. We use the correspondence between lagrangian submanifolds of Gr(1, n) and lagrangian submanifolds of variety M_{n-2}, given by symplectic reduction Gr(1, n)//T^2 for some specially chosen moment maps, which generate T^2 action on Gr(1, n). We establish that in this way one finds many topological types, realized by lagrangian submanifolds, and then one counts that Gr(1, n) admits more than n different topological types of smooth lagrangian submanifolds. | math | 0 |
Title: HEAP: Unsupervised Object Discovery and Localization with Contrastive Grouping
Abstract: Unsupervised object discovery and localization aims to detect or segment objects in an image without any supervision. Recent efforts have demonstrated a notable potential to identify salient foreground objects by utilizing self-supervised transformer features. However, their scopes only build upon patch-level features within an image, neglecting region/image-level and cross-image relationships at a broader scale. Moreover, these methods cannot differentiate various semantics from multiple instances. To address these problems, we introduce Hierarchical mErging framework via contrAstive grouPing (HEAP). Specifically, a novel lightweight head with cross-attention mechanism is designed to adaptively group intra-image patches into semantically coherent regions based on correlation among self-supervised features. Further, to ensure the distinguishability among various regions, we introduce a region-level contrastive clustering loss to pull closer similar regions across images. Also, an image-level contrastive loss is present to push foreground and background representations apart, with which foreground objects and background are accordingly discovered. HEAP facilitates efficient hierarchical image decomposition, which contributes to more accurate object discovery while also enabling differentiation among objects of various classes. Extensive experimental results on semantic segmentation retrieval, unsupervised object discovery, and saliency detection tasks demonstrate that HEAP achieves state-of-the-art performance. | cs | 0 |
Title: AERIAL-CORE: AI-Powered Aerial Robots for Inspection and Maintenance of Electrical Power Infrastructures
Abstract: Large-scale infrastructures are prone to deterioration due to age, environmental influences, and heavy usage. Ensuring their safety through regular inspections and maintenance is crucial to prevent incidents that can significantly affect public safety and the environment. This is especially pertinent in the context of electrical power networks, which, while essential for energy provision, can also be sources of forest fires. Intelligent drones have the potential to revolutionize inspection and maintenance, eliminating the risks for human operators, increasing productivity, reducing inspection time, and improving data collection quality. However, most of the current methods and technologies in aerial robotics have been trialed primarily in indoor testbeds or outdoor settings under strictly controlled conditions, always within the line of sight of human operators. Additionally, these methods and technologies have typically been evaluated in isolation, lacking comprehensive integration. This paper introduces the first autonomous system that combines various innovative aerial robots. This system is designed for extended-range inspections beyond the visual line of sight, features aerial manipulators for maintenance tasks, and includes support mechanisms for human operators working at elevated heights. The paper further discusses the successful validation of this system on numerous electrical power lines, with aerial robots executing flights over 10 kilometers away from their ground control stations. | cs | 0 |