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A Polynomial Time Iterative Algorithm for Matching Gaussian Matrices with Non-vanishing Correlation 匹配非相关性高斯矩阵的多项式时间迭代算法
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-22 DOI: 10.1007/s10208-024-09662-x
Jian Ding, Zhangsong Li
{"title":"A Polynomial Time Iterative Algorithm for Matching Gaussian Matrices with Non-vanishing Correlation","authors":"Jian Ding, Zhangsong Li","doi":"10.1007/s10208-024-09662-x","DOIUrl":"https://doi.org/10.1007/s10208-024-09662-x","url":null,"abstract":"<p>Motivated by the problem of matching vertices in two correlated Erdős-Rényi graphs, we study the problem of matching two correlated Gaussian Wigner matrices. We propose an iterative matching algorithm, which succeeds in polynomial time as long as the correlation between the two Gaussian matrices does not vanish. Our result is the first polynomial time algorithm that solves a graph matching type of problem when the correlation is an arbitrarily small constant.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantitative Stability of the Pushforward Operation by an Optimal Transport Map 通过最优传输图实现前推操作的定量稳定性
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-19 DOI: 10.1007/s10208-024-09669-4
Guillaume Carlier, Alex Delalande, Quentin Mérigot
{"title":"Quantitative Stability of the Pushforward Operation by an Optimal Transport Map","authors":"Guillaume Carlier, Alex Delalande, Quentin Mérigot","doi":"10.1007/s10208-024-09669-4","DOIUrl":"https://doi.org/10.1007/s10208-024-09669-4","url":null,"abstract":"<p>We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141730631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Regularized Stein Variational Gradient Flow 正则化斯坦因变异梯度流
IF 2.5 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-17 DOI: 10.1007/s10208-024-09663-w
Ye He, Krishnakumar Balasubramanian, Bharath K. Sriperumbudur, Jianfeng Lu
{"title":"Regularized Stein Variational Gradient Flow","authors":"Ye He, Krishnakumar Balasubramanian, Bharath K. Sriperumbudur, Jianfeng Lu","doi":"10.1007/s10208-024-09663-w","DOIUrl":"https://doi.org/10.1007/s10208-024-09663-w","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141830776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Correction to: Lie–Poisson Methods for Isospectral Flows 更正：等谱流动的列-泊松方法
IF 2.5 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-09 DOI: 10.1007/s10208-024-09661-y
K. Modin, M. Viviani
{"title":"Correction to: Lie–Poisson Methods for Isospectral Flows","authors":"K. Modin, M. Viviani","doi":"10.1007/s10208-024-09661-y","DOIUrl":"https://doi.org/10.1007/s10208-024-09661-y","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141665086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Koszul Complexes and Relative Homological Algebra of Functors Over Posets Koszul 复数和 Posets 上函数的相对同调代数
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-06-18 DOI: 10.1007/s10208-024-09660-z
Wojciech Chachólski, Andrea Guidolin, Isaac Ren, Martina Scolamiero, Francesca Tombari
{"title":"Koszul Complexes and Relative Homological Algebra of Functors Over Posets","authors":"Wojciech Chachólski, Andrea Guidolin, Isaac Ren, Martina Scolamiero, Francesca Tombari","doi":"10.1007/s10208-024-09660-z","DOIUrl":"https://doi.org/10.1007/s10208-024-09660-z","url":null,"abstract":"<p>Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141425513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Local Nearly Linearly Convergent First-Order Method for Nonsmooth Functions with Quadratic Growth 具有二次增长的非光滑函数的局部近线性收敛一阶方法
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-06-14 DOI: 10.1007/s10208-024-09653-y
Damek Davis, Liwei Jiang
{"title":"A Local Nearly Linearly Convergent First-Order Method for Nonsmooth Functions with Quadratic Growth","authors":"Damek Davis, Liwei Jiang","doi":"10.1007/s10208-024-09653-y","DOIUrl":"https://doi.org/10.1007/s10208-024-09653-y","url":null,"abstract":"<p>Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro’s decomposable class, and generic semialgebraic functions. The algorithm is parameter-free and derives from Goldstein’s conceptual subgradient method.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141326869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers 逆问题中的收敛正则化和线性即插即用去噪器
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-06-03 DOI: 10.1007/s10208-024-09654-x
Andreas Hauptmann, Subhadip Mukherjee, Carola-Bibiane Schönlieb, Ferdia Sherry
{"title":"Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers","authors":"Andreas Hauptmann, Subhadip Mukherjee, Carola-Bibiane Schönlieb, Ferdia Sherry","doi":"10.1007/s10208-024-09654-x","DOIUrl":"https://doi.org/10.1007/s10208-024-09654-x","url":null,"abstract":"<p>Regularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, thatis under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141246310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Identifiability, the KL Property in Metric Spaces, and Subgradient Curves 可识别性、公度空间中的 KL 特性和次梯度曲线
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-05-28 DOI: 10.1007/s10208-024-09652-z
A. S. Lewis, Tonghua Tian
{"title":"Identifiability, the KL Property in Metric Spaces, and Subgradient Curves","authors":"A. S. Lewis, Tonghua Tian","doi":"10.1007/s10208-024-09652-z","DOIUrl":"https://doi.org/10.1007/s10208-024-09652-z","url":null,"abstract":"<p>Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves. The Kurdyka–Łojasiewicz property typically governs convergence in both discrete and continuous time: we explore its interplay with identifiability.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimal Approximation of Unique Continuation 唯一连续性的最佳近似值
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-05-20 DOI: 10.1007/s10208-024-09655-w
Erik Burman, Mihai Nechita, Lauri Oksanen
{"title":"Optimal Approximation of Unique Continuation","authors":"Erik Burman, Mihai Nechita, Lauri Oksanen","doi":"10.1007/s10208-024-09655-w","DOIUrl":"https://doi.org/10.1007/s10208-024-09655-w","url":null,"abstract":"<p>We consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of <i>optimal error estimates</i> is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141074297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Group-Invariant Max Filtering 组不变最大过滤
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-05-17 DOI: 10.1007/s10208-024-09656-9
Jameson Cahill, Joseph W. Iverson, Dustin G. Mixon, Daniel Packer
{"title":"Group-Invariant Max Filtering","authors":"Jameson Cahill, Joseph W. Iverson, Dustin G. Mixon, Daniel Packer","doi":"10.1007/s10208-024-09656-9","DOIUrl":"https://doi.org/10.1007/s10208-024-09656-9","url":null,"abstract":"<p>Given a real inner product space <i>V</i> and a group <i>G</i> of linear isometries, we construct a family of <i>G</i>-invariant real-valued functions on <i>V</i> that we call <i>max filters</i>. In the case where <span>(V={mathbb {R}}^d)</span> and <i>G</i> is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where <span>(V=L^2({mathbb {R}}^d))</span> and <i>G</i> is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140953985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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