Applied and Computational Harmonic Analysis最新文献

{"title":"Frame set for shifted sinc-function","authors":"Yurii Belov ,&nbsp;Andrei V. Semenov","doi":"10.1016/j.acha.2024.101654","DOIUrl":"10.1016/j.acha.2024.101654","url":null,"abstract":"<div><p>We prove that frame set <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> for imaginary shift of sinc-function<span><span><span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mi>π</mi><mi>b</mi><mo>(</mo><mi>t</mi><mo>−</mo><mi>i</mi><mi>w</mi><mo>)</mo></mrow><mrow><mi>t</mi><mo>−</mo><mi>i</mi><mi>w</mi></mrow></mfrac><mo>,</mo><mspace></mspace><mi>b</mi><mo>,</mo><mi>w</mi><mo>∈</mo><mi>R</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span></span></span> can be described as <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>:</mo><mi>α</mi><mi>β</mi><mo>⩽</mo><mn>1</mn><mo>,</mo><mi>β</mi><mo>⩽</mo><mo>|</mo><mi>b</mi><mo>|</mo><mo>}</mo></math></span>.</p><p>In addition, we prove that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>:</mo><mi>α</mi><mi>β</mi><mo>⩽</mo><mn>1</mn><mo>}</mo></math></span> for window functions <em>g</em> of the form<span><span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mi>i</mi><mi>w</mi></mrow></mfrac><mo>(</mo><mn>1</mn><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>i</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>t</mi></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span> such that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>⩾</mo><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>π</mi><mo>|</mo><mi>w</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo></mrow></msup><mo>&lt;</mo><mn>1</mn></math></span>, <span><math><mi>w</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>&lt;</mo><mn>0</mn></math></span>.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101654"},"PeriodicalIF":2.5,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140182455","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}

{"title":"Eigenmatrix for unstructured sparse recovery","authors":"Lexing Ying","doi":"10.1016/j.acha.2024.101653","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101653","url":null,"abstract":"<div><p>This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101653"},"PeriodicalIF":2.5,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140133908","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}

{"title":"Solving PDEs on unknown manifolds with machine learning","authors":"Senwei Liang ,&nbsp;Shixiao W. Jiang ,&nbsp;John Harlim ,&nbsp;Haizhao Yang","doi":"10.1016/j.acha.2024.101652","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101652","url":null,"abstract":"<div><p>This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nyström-based interpolation method.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101652"},"PeriodicalIF":2.5,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140014304","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}

{"title":"Separation-free spectral super-resolution via convex optimization","authors":"Zai Yang ,&nbsp;Yi-Lin Mo ,&nbsp;Zongben Xu","doi":"10.1016/j.acha.2024.101650","DOIUrl":"10.1016/j.acha.2024.101650","url":null,"abstract":"<div><p>Atomic norm methods have recently been proposed for spectral super-resolution with flexibility in dealing with missing data and miscellaneous noises. A notorious drawback of these convex optimization methods however is their lower resolution in the high signal-to-noise (SNR) regime as compared to conventional methods such as ESPRIT. In this paper, we devise a simple weighting scheme in existing atomic norm methods and show that in theory the resolution of the resulting convex optimization method can be made arbitrarily high in the absence of noise, achieving the so-called separation-free super-resolution. This is proved by a novel, kernel-free construction of the dual certificate whose existence guarantees exact super-resolution using the proposed method. Numerical results corroborating our analysis are provided.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101650"},"PeriodicalIF":2.5,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043807","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}

{"title":"Marcinkiewicz–Zygmund inequalities for scattered and random data on the q-sphere","authors":"Frank Filbir ,&nbsp;Ralf Hielscher ,&nbsp;Thomas Jahn ,&nbsp;Tino Ullrich","doi":"10.1016/j.acha.2024.101651","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101651","url":null,"abstract":"<div><p>The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz–Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the <em>q</em>-dimensional sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>, and investigate how well continuous <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-norms of polynomials <em>f</em> of maximum degree <em>n</em> on the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> can be discretized by positively weighted <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>, the dimension <em>q</em>, and the degree <em>n</em> of the polynomials.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101651"},"PeriodicalIF":2.5,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000289/pdfft?md5=c98b0bf5b8b162d91ccc058130ea9e34&pid=1-s2.0-S1063520324000289-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140042516","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}

{"title":"Exponential lower bound for the eigenvalues of the time-frequency localization operator before the plunge region","authors":"Aleksei Kulikov","doi":"10.1016/j.acha.2024.101639","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101639","url":null,"abstract":"<div><p>For a pair of sets <span><math><mi>T</mi><mo>,</mo><mi>Ω</mi><mo>⊂</mo><mi>R</mi></math></span> the time-frequency localization operator is defined as <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi><mo>,</mo><mi>Ω</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>T</mi></mrow></msub><msup><mrow><mi>F</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>P</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mi>F</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span>, where <span><math><mi>F</mi></math></span> is the Fourier transform and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span> are projection operators onto <em>T</em> and Ω, respectively. We show that in the case when both <em>T</em> and Ω are intervals, the eigenvalues of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi><mo>,</mo><mi>Ω</mi></mrow></msub></math></span> satisfy <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>,</mo><mi>Ω</mi><mo>)</mo><mo>≥</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>δ</mi></mrow><mrow><mo>|</mo><mi>T</mi><mo>|</mo><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></msup></math></span> if <span><math><mi>n</mi><mo>≤</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mo>|</mo><mi>T</mi><mo>|</mo><mo>|</mo><mi>Ω</mi><mo>|</mo></math></span>, where <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> is arbitrary and <span><math><mi>δ</mi><mo>=</mo><mi>δ</mi><mo>(</mo><mi>ε</mi><mo>)</mo><mo>&lt;</mo><mn>1</mn></math></span>, provided that <span><math><mo>|</mo><mi>T</mi><mo>|</mo><mo>|</mo><mi>Ω</mi><mo>|</mo><mo>&gt;</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>ε</mi></mrow></msub></math></span>. This improves the result of Bonami, Jaming and Karoui, who proved it for <span><math><mi>ε</mi><mo>≥</mo><mn>0.42</mn></math></span>. The proof is based on the properties of the Bargmann transform.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101639"},"PeriodicalIF":2.5,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139999730","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}

{"title":"Uniform approximation of common Gaussian process kernels using equispaced Fourier grids","authors":"Alex Barnett ,&nbsp;Philip Greengard ,&nbsp;Manas Rachh","doi":"10.1016/j.acha.2024.101640","DOIUrl":"10.1016/j.acha.2024.101640","url":null,"abstract":"<div><p>The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101640"},"PeriodicalIF":2.5,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139994414","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}

{"title":"Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold","authors":"Vlado Menkovski ,&nbsp;Jacobus W. Portegies ,&nbsp;Mahefa Ratsisetraina Ravelonanosy","doi":"10.1016/j.acha.2024.101642","DOIUrl":"10.1016/j.acha.2024.101642","url":null,"abstract":"<div><p>We give an asymptotic expansion of the relative entropy between the heat kernel <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> of a compact Riemannian manifold <em>Z</em> and the normalized Riemannian volume for small values of <em>t</em> and for a fixed element <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>. We prove that coefficients in the expansion can be expressed as universal polynomials in the components of the curvature tensor and its covariant derivatives at <em>z</em>, when they are expressed in terms of normal coordinates. We describe a method to compute the coefficients, and we use the method to compute the first three coefficients. The asymptotic expansion is necessary for an unsupervised machine-learning algorithm called the Diffusion Variational Autoencoder.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101642"},"PeriodicalIF":2.5,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000198/pdfft?md5=9b07347114acdc753144d27860b6f702&pid=1-s2.0-S1063520324000198-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937845","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}

{"title":"Variable bandwidth via Wilson bases","authors":"Beatrice Andreolli,&nbsp;Karlheinz Gröchenig","doi":"10.1016/j.acha.2024.101641","DOIUrl":"10.1016/j.acha.2024.101641","url":null,"abstract":"<div><p>We introduce a new concept of variable bandwidth that is based on the frequency truncation of Wilson expansions. For this model we derive sampling theorems, a complete reconstruction of <em>f</em> from its samples, and necessary density conditions for sampling. Numerical simulations support the interpretation of this model of variable bandwidth. In particular, chirps, as they arise in the description of gravitational waves, can be modeled in a space of variable bandwidth.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101641"},"PeriodicalIF":2.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000186/pdfft?md5=a1bc8edd6739aca166f773e9d3ff503a&pid=1-s2.0-S1063520324000186-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937837","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}

{"title":"The G-invariant graph Laplacian Part I: Convergence rate and eigendecomposition","authors":"Eitan Rosen ,&nbsp;Paulina Hoyos ,&nbsp;Xiuyuan Cheng ,&nbsp;Joe Kileel ,&nbsp;Yoel Shkolnisky","doi":"10.1016/j.acha.2024.101637","DOIUrl":"10.1016/j.acha.2024.101637","url":null,"abstract":"<div><p>Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group <em>G</em>. We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of <em>G</em> on the data set. We deem the latter construction the “<em>G</em>-invariant Graph Laplacian” (<em>G</em>-GL). We show that the <em>G</em>-GL converges to the Laplace-Beltrami operator on the data manifold, while enjoying a significantly improved convergence rate compared to the standard graph Laplacian which only utilizes the distances between the points in the given data set. Furthermore, we show that the <em>G</em>-GL admits a set of eigenfunctions that have the form of certain products between the group elements and eigenvectors of certain matrices, which can be estimated from the data efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group <span><math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101637"},"PeriodicalIF":2.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937829","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}