Applied and Computational Harmonic Analysis最新文献_第3页

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions 拉普拉斯-贝尔特拉米特征函数中更高的特征切格比

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-21 DOI: 10.1016/j.acha.2024.101710

Gary Froyland, Christopher P. Rock

{"title":"Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions","authors":"Gary Froyland, Christopher P. Rock","doi":"10.1016/j.acha.2024.101710","DOIUrl":"10.1016/j.acha.2024.101710","url":null,"abstract":"<div><div>This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.</div><div>Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101710"},"PeriodicalIF":2.6,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A perturbative analysis for noisy spectral estimation 噪声频谱估计的扰动分析

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-18 DOI: 10.1016/j.acha.2024.101716

Lexing Ying

引用次数: 0

Solving PDEs on spheres with physics-informed convolutional neural networks 用物理信息卷积神经网络求解球面上的 PDEs

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-15 DOI: 10.1016/j.acha.2024.101714

Guanhang Lei , Zhen Lei , Lei Shi , Chenyu Zeng , Ding-Xuan Zhou

{"title":"Solving PDEs on spheres with physics-informed convolutional neural networks","authors":"Guanhang Lei , Zhen Lei , Lei Shi , Chenyu Zeng , Ding-Xuan Zhou","doi":"10.1016/j.acha.2024.101714","DOIUrl":"10.1016/j.acha.2024.101714","url":null,"abstract":"<div><div>Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101714"},"PeriodicalIF":2.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Linearized Wasserstein dimensionality reduction with approximation guarantees 具有近似保证的线性化瓦瑟斯坦降维法

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-15 DOI: 10.1016/j.acha.2024.101718

Alexander Cloninger , Keaton Hamm , Varun Khurana , Caroline Moosmüller

{"title":"Linearized Wasserstein dimensionality reduction with approximation guarantees","authors":"Alexander Cloninger , Keaton Hamm , Varun Khurana , Caroline Moosmüller","doi":"10.1016/j.acha.2024.101718","DOIUrl":"10.1016/j.acha.2024.101718","url":null,"abstract":"<div><div>We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101718"},"PeriodicalIF":2.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Weighted variation spaces and approximation by shallow ReLU networks 加权变异空间和浅层 ReLU 网络逼近

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-10 DOI: 10.1016/j.acha.2024.101713

Ronald DeVore , Robert D. Nowak , Rahul Parhi , Jonathan W. Siegel

{"title":"Weighted variation spaces and approximation by shallow ReLU networks","authors":"Ronald DeVore , Robert D. Nowak , Rahul Parhi , Jonathan W. Siegel","doi":"10.1016/j.acha.2024.101713","DOIUrl":"10.1016/j.acha.2024.101713","url":null,"abstract":"<div><div>We investigate the approximation of functions <em>f</em> on a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> by the outputs of single-hidden-layer ReLU neural networks of width <em>n</em>. This form of nonlinear <em>n</em>-term dictionary approximation has been intensely studied since it is the simplest case of neural network approximation (NNA). There are several celebrated approximation results for this form of NNA that introduce novel model classes of functions on Ω whose approximation rates do not grow unbounded with the input dimension. These novel classes include Barron classes, and classes based on sparsity or variation such as the Radon-domain BV classes. The present paper is concerned with the definition of these novel model classes on domains Ω. The current definition of these model classes does not depend on the domain Ω. A new and more proper definition of model classes on domains is given by introducing the concept of weighted variation spaces. These new model classes are intrinsic to the domain itself. The importance of these new model classes is that they are strictly larger than the classical (domain-independent) classes. Yet, it is shown that they maintain the same NNA rates.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101713"},"PeriodicalIF":2.6,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Two subspace methods for frequency sparse graph signals 频率稀疏图信号的两种子空间方法

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-02 DOI: 10.1016/j.acha.2024.101711

Tarek Emmrich, Martina Juhnke, Stefan Kunis

引用次数: 0

Approximation theory of wavelet frame based image restoration 基于小波帧的图像修复近似理论

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-09-27 DOI: 10.1016/j.acha.2024.101712

Jian-Feng Cai , Jae Kyu Choi , Jianbin Yang

引用次数: 0

Donoho-Logan large sieve principles for the wavelet transform 小波变换的 Donoho-Logan 大筛原理

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-09-26 DOI: 10.1016/j.acha.2024.101709

Luís Daniel Abreu , Michael Speckbacher

引用次数: 0

Data-driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application 具有可分离协方差结构的高维噪声下奇异值的数据驱动优化收缩及其应用

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-09-12 DOI: 10.1016/j.acha.2024.101698

Pei-Chun Su , Hau-Tieng Wu

{"title":"Data-driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application","authors":"Pei-Chun Su , Hau-Tieng Wu","doi":"10.1016/j.acha.2024.101698","DOIUrl":"10.1016/j.acha.2024.101698","url":null,"abstract":"<div><p>We develop a data-driven optimal shrinkage algorithm, named <em>extended OptShrink</em> (eOptShrink), for matrix denoising with high-dimensional noise and a separable covariance structure. This noise is colored and dependent across samples. The algorithm leverages the asymptotic behavior of singular values and vectors of the noisy data's random matrix. Our theory includes the sticking property of non-outlier singular values, delocalization of weak signal singular vectors, and the spectral behavior of outlier singular values and vectors. We introduce three estimators: a novel rank estimator, an estimator for the spectral distribution of the pure noise matrix, and the optimal shrinker eOptShrink. Notably, eOptShrink does not require estimating the noise's separable covariance structure. We provide a theoretical guarantee for these estimators with a convergence rate. Through numerical simulations and comparisons with state-of-the-art optimal shrinkage algorithms, we demonstrate eOptShrink's application in extracting maternal and fetal electrocardiograms from single-channel trans-abdominal maternal electrocardiograms.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101698"},"PeriodicalIF":2.6,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups 构建由 LCA 组滤波器生成的成对正交 Parseval 框架

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-09-07 DOI: 10.1016/j.acha.2024.101708

Navneet Redhu , Anupam Gumber , Niraj K. Shukla

{"title":"Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups","authors":"Navneet Redhu , Anupam Gumber , Niraj K. Shukla","doi":"10.1016/j.acha.2024.101708","DOIUrl":"10.1016/j.acha.2024.101708","url":null,"abstract":"<div><p>The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfying the local integrability condition (LIC) and having the Calderón sum one, where <em>G</em> is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing <em>N</em> numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101708"},"PeriodicalIF":2.6,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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