Applied and Computational Harmonic Analysis最新文献_第9页

Exponential bases for partitions of intervals 区间划分的指数基

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-29 DOI: 10.1016/j.acha.2023.101607

Götz Pfander , Shauna Revay , David Walnut

引用次数: 1

A multivariate Riesz basis of ReLU neural networks ReLU神经网络的多元Riesz基

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-20 DOI: 10.1016/j.acha.2023.101605

Cornelia Schneider , Jan Vybíral

引用次数: 0

On the relation between Fourier and Walsh–Rademacher spectra for random fields 关于随机场的傅立叶谱和Walsh–Rademacher谱之间的关系

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-13 DOI: 10.1016/j.acha.2023.101603

Anton Kutsenko , Sergey Danilov , Stephan Juricke , Marcel Oliver

引用次数: 0

Deep nonparametric estimation of intrinsic data structures by chart autoencoders: Generalization error and robustness 图表自动编码器对内在数据结构的深度非参数估计：泛化误差和稳健性

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-12 DOI: 10.1016/j.acha.2023.101602

Hao Liu , Alex Havrilla , Rongjie Lai , Wenjing Liao

{"title":"Deep nonparametric estimation of intrinsic data structures by chart autoencoders: Generalization error and robustness","authors":"Hao Liu , Alex Havrilla , Rongjie Lai , Wenjing Liao","doi":"10.1016/j.acha.2023.101602","DOIUrl":"https://doi.org/10.1016/j.acha.2023.101602","url":null,"abstract":"<div>Autoencoders have demonstrated remarkable success in learning low-dimensional latent features of high-dimensional data across various applications. Assuming that data are sampled near a low-dimensional manifold, we employ chart autoencoders, which encode data into low-dimensional latent features on a collection of charts, preserving the topology and geometry of the data manifold. Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering n noisy training samples, along with their noise-free counterparts, on a d-dimensional manifold. By training autoencoders, we show that chart autoencoders can effectively denoise the input data with normal noise. We prove that, under proper network architectures, chart autoencoders achieve a squared generalization error in the order of <math><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>⁡</mo><mi>n</mi></math>, which depends on the intrinsic dimension of the manifold and only weakly depends on the ambient dimension and noise level. We further extend our theory on data with noise containing both normal and tangential components, where chart autoencoders still exhibit a denoising effect for the normal component. As a special case, our theory also applies to classical autoencoders, as long as the data manifold has a global parametrization. Our results provide a solid theoretical foundation for the effectiveness of autoencoders, which is further validated through several numerical experiments.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101602"},"PeriodicalIF":2.5,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49778386","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

Time and band limiting for exceptional polynomials 例外多项式的时间和频带限制

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-10 DOI: 10.1016/j.acha.2023.101600

M.M. Castro , F.A. Grünbaum , I. Zurrián

引用次数: 0

LU decomposition and Toeplitz decomposition of a neural network 神经网络的LU分解和Toeplitz分解

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-06 DOI: 10.1016/j.acha.2023.101601

Yucong Liu , Simiao Jiao , Lek-Heng Lim

引用次数: 0

Representation of operators using fusion frames 使用融合框架表示运算符

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-10-05 DOI: 10.1016/j.acha.2023.101596

Peter Balazs, Mitra Shamsabadi, Ali Akbar Arefijamaal, Gilles Chardon

{"title":"Representation of operators using fusion frames","authors":"Peter Balazs, Mitra Shamsabadi, Ali Akbar Arefijamaal, Gilles Chardon","doi":"10.1016/j.acha.2023.101596","DOIUrl":"https://doi.org/10.1016/j.acha.2023.101596","url":null,"abstract":"<div>To solve operator equations numerically, matrix representations are employing bases or more recently frames. For finding the numerical solution of operator equations a decomposition in subspaces is needed in many applications. To combine those two approaches, it is necessary to extend the known methods of matrix representation to the utilization of fusion frames. In this paper, we investigate this representation of operators on a Hilbert space <math><mi>H</mi></math> with Bessel fusion sequences, fusion frames and fusion Riesz bases. Fusion frames can be considered as a frame-like family of subspaces. Taking the particular property of the duality of fusion frames into account, this allows us to define a matrix representation in a canonical as well as an alternate way, the later being more efficient and well behaved in respect to inversion. We will give the basic definitions and show some structural results, like that the functions assigning the alternate representation to an operator is an algebra homomorphism. We give formulas for pseudo-inverses and the inverses (if existing) of such matrix representations. We apply this idea to Schatten p-class operators. Consequently, we show that tensor products of fusion frames are fusion frames in the space of Hilbert-Schmidt operators. We will show how this can be used for the solution of operator equations and link our approach to the additive Schwarz algorithm. Consequently, we propose some methods for solving an operator equation by iterative methods on the subspaces. Moreover, we implement the alternate Schwarz algorithms employing our perspective and provide small proof-of-concept numerical experiments. Finally, we show the application of this concept to overlapped convolution and the non-standard wavelet representation of operators.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101596"},"PeriodicalIF":2.5,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49778388","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: 3

Diffusion maps for embedded manifolds with boundary with applications to PDEs 具有边界的嵌入流形的扩散映射及其在偏微分方程中的应用

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-09-09 DOI: 10.1016/j.acha.2023.101593

Ryan Vaughn , Tyrus Berry , Harbir Antil

{"title":"Diffusion maps for embedded manifolds with boundary with applications to PDEs","authors":"Ryan Vaughn , Tyrus Berry , Harbir Antil","doi":"10.1016/j.acha.2023.101593","DOIUrl":"https://doi.org/10.1016/j.acha.2023.101593","url":null,"abstract":"<div>Given only a finite collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to numerically solve elliptic and parabolic partial differential equations (PDEs) supplemented with boundary conditions. Since the construction of triangulations on unknown manifolds can be both difficult and expensive, both in terms of computational and data requirements, our goal is to solve these problems without a triangulation. Instead, we rely only on using the sample points to define quadrature formulas on the unknown manifold. Our main tool is the diffusion maps algorithm. We re-analyze this well-known method in a variational sense for manifolds with boundary. Our main result is that the variational diffusion maps graph Laplacian is a consistent estimator of the Dirichlet energy on the manifold. This improves upon previous results and provides a rigorous justification of the well-known relationship between diffusion maps and the Neumann eigenvalue problem. Moreover, using semigeodesic coordinates we derive the first uniform asymptotic expansion of the diffusion maps kernel integral operator for manifolds with boundary. This expansion relies on a novel lemma which relates the extrinsic Euclidean distance to the coordinate norm in a normal collar of the boundary. We then use a recently developed method of estimating the distance to boundary function (notice that the boundary location is assumed to be unknown) to construct a consistent estimator for boundary integrals. Finally, by combining these various estimators, we illustrate how to impose Dirichlet and Neumann conditions for some common PDEs based on the Laplacian. Several numerical examples illustrate our theoretical findings.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101593"},"PeriodicalIF":2.5,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49778390","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: 14

Metaplectic Gabor frames and symplectic analysis of time-frequency spaces 复复Gabor框架与时频空间的辛分析

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-09-09 DOI: 10.1016/j.acha.2023.101594

Elena Cordero , Gianluca Giacchi

引用次数: 2

On generalizations of the nonwindowed scattering transform 关于无窗散射变换的推广。

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-09-09 DOI: 10.1016/j.acha.2023.101597

Albert Chua , Matthew Hirn , Anna Little

引用次数: 0