Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a growing interest in defining transport-based distances that allow for comparing signed measures and, more generally, multi-channeled signals. Transport $\mathrm{L}^{p}$ distances are notable extensions of the optimal transport framework to signed and possibly multi-channeled signals. In this paper, we introduce partial transport $\mathrm{L}^{p}$ distances as a new family of metrics for comparing generic signals, benefiting from the robustness of partial transport distances. We provide theoretical background such as the existence of optimal plans and the behavior of the distance in various limits. Furthermore, we introduce the sliced variation of these distances, which allows for rapid comparison of generic signals. Finally, we demonstrate the application of the proposed distances in signal class separability and nearest neighbor classification.
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Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Card-based protocols to compute various Boolean functions have been developed. As each input bit is typically encoded by two cards, computing an $n$-variable Boolean function requires at least $2n$ cards. We are interested in optimal protocols that use exactly $2n$ cards. In particular, we focus on symmetric functions. In this paper, we formulate the problem of developing $2n$-card protocols to compute $n$-variable symmetric Boolean functions by classifying all such functions into several NPN-equivalence classes. We then summarize existing protocols that can compute some representative functions from these classes, and also solve some open problems in the cases $n=4$, 5, 6, and 7. In particular, we develop a protocol to compute a function $k$Mod3, which determines whether the sum of all inputs is congruent to $k$ modulo 3 ($k \in \{0,1,2\}$).
Most existing salient object detection methods mostly use U-Net or feature pyramid structure, which simply aggregates feature maps of different scales, ignoring the uniqueness and interdependence of them and their respective contributions to the final prediction. To overcome these, we propose the M$^3$Net, i.e., the Multilevel, Mixed and Multistage attention network for Salient Object Detection (SOD). Firstly, we propose Multiscale Interaction Block which innovatively introduces the cross-attention approach to achieve the interaction between multilevel features, allowing high-level features to guide low-level feature learning and thus enhancing salient regions. Secondly, considering the fact that previous Transformer based SOD methods locate salient regions only using global self-attention while inevitably overlooking the details of complex objects, we propose the Mixed Attention Block. This block combines global self-attention and window self-attention, aiming at modeling context at both global and local levels to further improve the accuracy of the prediction map. Finally, we proposed a multilevel supervision strategy to optimize the aggregated feature stage-by-stage. Experiments on six challenging datasets demonstrate that the proposed M$^3$Net surpasses recent CNN and Transformer-based SOD arts in terms of four metrics. Codes are available at //github.com/I2-Multimedia-Lab/M3Net.
The field of category theory seeks to unify and generalize concepts and constructions across different areas of mathematics, from algebra to geometry to topology and also to logic and theoretical computer science. Formalized $1$-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure", rely on infinite-dimensional categories in place of $1$-dimensional categories, and $\infty$-category theory has thus far proved unamenable to computer formalization. Using a new proof assistant called Rzk, which is designed to support Riehl-Shulman's simplicial extension of homotopy type theory for synthetic $\infty$-category theory, we provide the first formalizations of results from $\infty$-category theory. This includes in particular a formalization of the Yoneda lemma, often regarded as the fundamental theorem of category theory, a theorem which roughly states that an object of a given category is determined by its relationship to all of the other objects of the category. A key feature of our framework is that, thanks to the synthetic theory, many constructions are automatically natural or functorial. We plan to use Rzk to formalize further results from $\infty$-category theory, such as the theory of limits and colimits and adjunctions.
We consider the problem of locating a nearest descriptor system of prescribed reduced order to a descriptor system with large order with respect to the ${\mathcal L}_\infty$ norm. Widely employed approaches such as the balanced truncation and best Hankel norm approximation for this ${\mathcal L}_\infty$ model reduction problem are usually expensive and yield solutions that are not optimal, not even locally. We propose approaches based on the minimization of the ${\mathcal L}_\infty$ objective by means of smooth optimization techniques. As we illustrate, direct applications of smooth optimization techniques are not feasible, since the optimization techniques converge at best at a linear rate requiring too many evaluations of the costly ${\mathcal L}_\infty$-norm objective to be practical. We replace the original large-scale system with a system of smaller order that interpolates the original system at points on the imaginary axis, and minimize the ${\mathcal L}_\infty$ objective after this replacement. The smaller system is refined by interpolating at additional imaginary points determined based on the local minimizer of the ${\mathcal L}_\infty$ objective, and the optimization is repeated. We argue the framework converges at a quadratic rate under smoothness and nondegeneracy assumptions, and describe how asymptotic stability constraints on the reduced system sought can be incorporated into our approach. The numerical experiments on benchmark examples illustrate that the approach leads to locally optimal solutions to the ${\mathcal L}_\infty$ model reduction problem, and the convergence occurs quickly for descriptors systems of order a few ten thousands.
A method is proposed for evaluation of single and double layer potentials of the Laplace and Helmholtz equations on piecewise smooth manifold boundary elements with constant densities. The method is based on a novel two-term decomposition of the layer potentials, derived by means of differential geometry. The first term is an integral of a differential 2-form which can be reduced to contour integrals using Stokes' theorem, while the second term is related to the element curvature. This decomposition reduces the degree of singularity and the curvature term can be further regularized by a polar coordinate transform. The method can handle singular and nearly singular integrals. Numerical results validating the accuracy of the method are presented for all combinations of single and double layer potentials, for the Laplace and Helmholtz kernels, and for singular and nearly singular integrals.
The Chisholm rational approximant is a natural generalization to two variables of the well-known single variable Pad\'e approximant, and has the advantage of reducing to the latter when one of the variables is set equals to 0. We present, to our knowledge, the first automated Mathematica package to evaluate diagonal Chisholm approximants of two variable series. For the moment, the package can only be used to evaluate diagonal approximants i.e. the maximum powers of both the variables, in both the numerator and the denominator, is equal to some integer $M$. We further modify the original method so as to allow us to evaluate the approximants around some general point $(x,y)$ not necessarily $(0,0)$. Using the approximants around general point $(x,y)$, allows us to get a better estimate of the result when the point of evaluation is far from $(0,0)$. Several examples of the elementary functions have been studied which shows that the approximants can be useful for analytic continuation and convergence acceleration purposes. We continue our study using various examples of two variable hypergeometric series, $\mathrm{Li}_{2,2}(x,y)$ etc that arise in particle physics and in the study of critical phenomena in condensed matter physics. The demonstration of the package is discussed in detail and the Mathematica package is provided as an ancillary file.
This work proposes $\texttt{NePhi}$, a neural deformation model which results in approximately diffeomorphic transformations. In contrast to the predominant voxel-based approaches, $\texttt{NePhi}$ represents deformations functionally which allows for memory-efficient training and inference. This is of particular importance for large volumetric registrations. Further, while medical image registration approaches representing transformation maps via multi-layer perceptrons have been proposed, $\texttt{NePhi}$ facilitates both pairwise optimization-based registration $\textit{as well as}$ learning-based registration via predicted or optimized global and local latent codes. Lastly, as deformation regularity is a highly desirable property for most medical image registration tasks, $\texttt{NePhi}$ makes use of gradient inverse consistency regularization which empirically results in approximately diffeomorphic transformations. We show the performance of $\texttt{NePhi}$ on two 2D synthetic datasets as well as on real 3D lung registration. Our results show that $\texttt{NePhi}$ can achieve similar accuracies as voxel-based representations in a single-resolution registration setting while using less memory and allowing for faster instance-optimization.
This study presents a comparative analysis of three predictive models with an increasing degree of flexibility: hidden dynamic geostatistical models (HDGM), generalised additive mixed models (GAMM), and the random forest spatiotemporal kriging models (RFSTK). These models are evaluated for their effectiveness in predicting PM$_{2.5}$ concentrations in Lombardy (North Italy) from 2016 to 2020. Despite differing methodologies, all models demonstrate proficient capture of spatiotemporal patterns within air pollution data with similar out-of-sample performance. Furthermore, the study delves into station-specific analyses, revealing variable model performance contingent on localised conditions. Model interpretation, facilitated by parametric coefficient analysis and partial dependence plots, unveils consistent associations between predictor variables and PM$_{2.5}$ concentrations. Despite nuanced variations in modelling spatiotemporal correlations, all models effectively accounted for the underlying dependence. In summary, this study underscores the efficacy of conventional techniques in modelling correlated spatiotemporal data, concurrently highlighting the complementary potential of Machine Learning and classical statistical approaches.
The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation $d=3(p^m-1)+1$ over $\mathrm{GF}(p^{2m})$ for any prime $p \ge 5$. Previously this problem was solved by Xia et al. only for $p=2$ and $p=3$ in a series of papers. The main difficulty of this problem for $p \ge 5$, as pointed out by Xia et al., is to count the number of codewords of "pure weight" 5 in $p$-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in $p$-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When $p \ge 7$, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.
Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at //github.com/Shen-Lab/L2-GCN.
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