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Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching

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arXiv 2022年

作者： Shi, Yunpeng Li, Shaohan Maunu, Tyler Lerman, Gilad Program in Applied and Computational Mathematics Princeton University United States School of Mathematics University of Minnesota United States Department of Mathematics Brandeis University United States

We develop new statistics for robustly filtering corrupted keypoint matches in the structure from motion pipeline. The statistics are based on consistency constraints that arise within the clustered structure of the graph of keypoint matches. The statistics are designed to give smaller values to corrupted matches and than uncorrupted matches. These new statistics are combined with an iterative reweighting scheme to filter keypoints, which can then be fed into any standard structure from motion pipeline. This filtering method can be efficiently implemented and scaled to massive datasets as it only requires sparse matrix multiplication. We demonstrate the efficacy of this method on synthetic and real structure from motion datasets and show that it achieves state-of-the-art accuracy and speed in these tasks. Copyright © 2022, The Authors. All rights reserved.

关键词： Statistics

Semiclassical Wkb Problem for the Non-Self-Adjoint Dirac Operator Withan Analytic Rapidly Oscillating Potential

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SSRN

SSRN 2022年

作者： Fujiie, Setsuro Hatzizisis, Nicholas Kamvissis, Spyridon Department of Mathematical Sciences Ritsumeikan University Japan Department of Mathematics & Applied Mathematics University of Crete Institute of Applied and Computational Mathematics FORTH Greece

In this paper we examine the semiclassical behaviorof the scattering data of anon-self-adjoint Dirac operator with a rapidly oscillatingpotential that is complex analytic in some neighborhood of the real *** of our results are rigorous and quite *** the other hand, complete and concrete understanding requires the investigation ofthe WKB geometry of specific *** such detailed computations we use a particular example that has been investigated numericallymore than 20 years ago by Bronski and Miller and rely heavily on their numerical *** employing the exact WKB method, we provide the complete rigorous uniformsemiclassical analysis of the Bohr-Sommerfeld condition for the location of theeigenvalues across unions of analytic arcs as well as the associated norming *** the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, weemploy instead an older theory that hasbeen developed in great detail by Olver. Our analysis is motivatedby the need to understand the semiclassical behaviour of the focusing cubic NLS equationwith initial data $A\exp\{iS/\epsilon\}$, in view of the well-known fact discoveredby Zakharov and Shabat that the spectral analysis of the Dirac operator enablesthe solution of the NLS equation via inverse scattering theory. © 2022, The Authors. All rights reserved.

关键词： Spectrum analysis

A SINGULAR TWO-PHASE STEFAN PROBLEM AND PARTICLES INTERACTING THROUGH THEIR HITTING TIMES

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arXiv 2022年

作者： Baker, Graeme Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance and Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel *** Codes 82C22, 60H30, 35K10, 35K60, 91G80 Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

EQUATIONS DRIVEN BY FAST-OSCILLATING FUNCTIONS OF A WIENER PROCESS

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arXiv 2023年

作者： Reese, Tanner M. Wehr, Jan Department of Mathematics University of Arizona TucsonAZ85721 United States Department of Mathematics and Program in Applied Mathematics University of Arizona TucsonAZ85721 United States

We study systems of differential equations driven by functions of a single Wiener process. In the limit of fast oscillations, we show that the solution process converges in law to the process defined by an SDE system driven by several independent Wiener processes. Drift terms appearing in the limiting equations can be interpreted as Stratonovich corrections. The problem has been motivated by experimental work and provides a rigorous treatment of equations arising from it. © 2023, CC0.

关键词： Differential equations

Semiclassical WKB Problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential

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arXiv 2022年

作者： Fujiié, Setsuro Hatzizisis, Nicholas Kamvissis, Spyridon Department of Mathematical Sciences Ritsumeikan University Japan Department of Mathematics & Applied Mathematics University of Crete Institute of Applied and Computational Mathematics FORTH Greece

In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues across unions of analytic arcs as well as the associated norming constants. For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data Aexp{iS/Ε}, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory. © 2022, CC BY.

关键词： Eigenvalues and eigenfunctions

A Multi-Dimensional Mathematical Model for Surface Exerted Point Forces in Elastic Media

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arXiv 2024年

作者： Asghar, Sabia Peng, Qiyao Vermolen, Fred J. Department of Mathematics and Statistics Computational Mathematics Group University of Hasselt Diepenbeek Belgium University of Hasselt Diepenbeek Belgium Mathematical Institute Leiden University Leiden Netherlands Department of Computer Science Vrije Universiteit Amsterdam Amsterdam Netherlands Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg South Africa Delft Institute of Applied Mathematics Delft University of Technology Mekelweg 4 Delft2628 CD Netherlands

Some phenotypes of biological cells exert mechanical forces on their direct environment during their development and progression. In this paper the impact of cellular forces on the surrounding tissue is considered. Assuming the size of the cell to be much smaller than that of the computational domain, and assuming small displacements, linear elasticity (Hooke’s Law) with point forces described by Dirac delta distributions is used in momentum balance equation. Due to the singular nature of the Dirac delta distribution, the solution does not lie in the classical H1 finite element space for multi-dimensional domains. We analyze the L2-convergence of forces in a superposition of line segments across the cell boundary to an integral representation of the forces on the cell boundary. It is proved that the L2-convergence of the displacement field away from the cell boundary matches the quadratic order of convergence of the midpoint rule on the forces that are exerted on the curve or surface that describes the cell boundary. © 2024, CC BY.

关键词： Cytology

CONVERGENCE OF CONTINUOUS NORMALIZING FLOWS FOR LEARNING PROBABILITY DISTRIBUTIONS

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arXiv 2024年

作者： Gao, By Yuan Huang, Jian Jiao, Yuling Zheng, Shurong Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong School of Mathematics and Statistics Hubei Key Laboratory of Computational Science Wuhan University Wuhan China School of Mathematics and Statistics Northeast Normal University Changchun China

Continuous normalizing flows (CNFs) are a generative method for learning probability distributions, which is based on ordinary differential equations. This method has shown remarkable empirical success across various applications, including large-scale image synthesis, protein structure prediction, and molecule generation. In this work, we study the theoretical properties of CNFs with linear interpolation in learning probability distributions from a finite random sample, using a flow matching objective function. We establish non-asymptotic error bounds for the distribution estimator based on CNFs, in terms of the Wasserstein-2 distance. The key assumption in our analysis is that the target distribution satisfies one of the following three conditions: it either has a bounded support, is strongly log-concave, or is a finite or infinite mixture of Gaussian distributions. We present a convergence analysis framework that encompasses the error due to velocity estimation, the discretization error, and the early stopping error. A key step in our analysis involves establishing the regularity properties of the velocity field and its estimator for CNFs constructed with linear interpolation. This necessitates the development of uniform error bounds with Lipschitz regularity control of deep ReLU networks that approximate the Lipschitz function class, which could be of independent interest. Our nonparametric convergence analysis offers theoretical guarantees for using CNFs to learn probability distributions from a finite random *** Codes 62G05, 68T07 © 2024, CC BY.

关键词： Ordinary differential equations

Robust Group Synchronization via Quadratic programming

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arXiv 2022年

作者： Shi, Yunpeng Wyeth, Cole Lerman, Gilad Program in Applied and Computational Mathematics Princeton University United States School of Mathematics University of Minnesota United States

We propose a novel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency of the group and we thus refer to our method as detection and estimation of structural consistency (DESC). This general framework can be extended to other algebraic and geometric structures. Our formulation has the following advantages: it can tolerate corruption as high as the information-theoretic bound, it does not require a good initialization for the estimates of group elements, it has a simple interpretation, and under some mild conditions the global minimum of our objective function exactly recovers the corruption levels. We demonstrate the competitive accuracy of our approach on both synthetic and real data experiments of rotation *** Codes 90C26, 90C17, 68Q87, 65C20, 90-08, 60-08 Copyright © 2022, The Authors. All rights reserved.

关键词： Quadratic programming

Energy Bounds for Discontinuous Galerkin Spectral Element Approximations of Well-Posed Overset Grid Problems for Hyperbolic Systems

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arXiv 2024年

作者： Kopriva, David A. Winters, Andrew R. Nordström, Jan Department of Mathematics The Florida State University TallahasseeFL32306 United States Computational Science Research Center San Diego State University San DiegoCA United States Department of Mathematics Applied Mathematics Linköping University Linköping581 83 Sweden Department of Mathematics and Applied Mathematics University of Johannesburg P.O. Box 524 Auckland Park2006 South Africa

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied. Copyright © 2024, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions