Foundations of Computational Mathematics最新文献_第7页

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

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

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":"47 1","pages":""},"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

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

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

引用次数: 0

A Sheaf-Theoretic Construction of Shape Space 形状空间的 Sheaf 理论构造

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-05-16 DOI: 10.1007/s10208-024-09650-1

Shreya Arya, Justin Curry, Sayan Mukherjee

引用次数: 0

Discrete Weber Inequalities and Related Maxwell Compactness for Hybrid Spaces over Polyhedral Partitions of Domains with General Topology 具有一般拓扑学的多面体分区域上混合空间的离散韦伯不等式及相关麦克斯韦紧凑性

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-04-16 DOI: 10.1007/s10208-024-09648-9

Simon Lemaire, Silvano Pitassi

引用次数: 0

Sum-of-Squares Relaxations for Information Theory and Variational Inference 信息论和变量推理的平方和松弛

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-04-05 DOI: 10.1007/s10208-024-09651-0

{"title":"Sum-of-Squares Relaxations for Information Theory and Variational Inference","authors":"","doi":"10.1007/s10208-024-09651-0","DOIUrl":"https://doi.org/10.1007/s10208-024-09651-0","url":null,"abstract":"<h3>Abstract</h3> <p>We consider extensions of the Shannon relative entropy, referred to as <em>f</em>-divergences. Three classical related computational problems are typically associated with these divergences: (a) estimation from moments, (b) computing normalizing integrals, and (c) variational inference in probabilistic models. These problems are related to one another through convex duality, and for all of them, there are many applications throughout data science, and we aim for computationally tractable approximation algorithms that preserve properties of the original problem such as potential convexity or monotonicity. In order to achieve this, we derive a sequence of convex relaxations for computing these divergences from non-centered covariance matrices associated with a given feature vector: starting from the typically non-tractable optimal lower-bound, we consider an additional relaxation based on “sums-of-squares”, which is is now computable in polynomial time as a semidefinite program. We also provide computationally more efficient relaxations based on spectral information divergences from quantum information theory. For all of the tasks above, beyond proposing new relaxations, we derive tractable convex optimization algorithms, and we present illustrations on multivariate trigonometric polynomials and functions on the Boolean hypercube.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"204 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140533954","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

Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs 求解高维和多尺度椭圆 PDE 的稀疏谱方法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-04-02 DOI: 10.1007/s10208-024-09649-8

{"title":"Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs","authors":"","doi":"10.1007/s10208-024-09649-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09649-8","url":null,"abstract":"<h3>Abstract</h3> <p>In his monograph <em>Chebyshev and Fourier Spectral Methods</em>, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, “[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues” [Boyd in Chebyshev and Fourier spectral methods, second rev ed. Dover Publications, Mineola, NY, 2001, pg. 194]. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, <em>sparse Fourier transform</em> paired with randomized rank-1 lattice methods. The resulting <em>sparse spectral method</em> rapidly and automatically determines a set of Fourier basis functions whose span is guaranteed to contain an accurate approximation of the solution of a given elliptic PDE. This much smaller, near-optimal Fourier basis is then used to efficiently solve the given PDE in a runtime which only depends on the PDE’s data compressibility and ellipticity properties, while breaking the curse of dimensionality and relieving linear dependence on any multiscale structure in the original problem. Theoretical performance of the method is established herein with convergence analysis in the Sobolev norm for a general class of non-constant diffusion equations, as well as pointers to technical extensions of the convergence analysis to more general advection–diffusion–reaction equations. Numerical experiments demonstrate good empirical performance on several multiscale and high-dimensional example problems, further showcasing the promise of the proposed methods in practice.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"68 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140343465","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