Applied and Computational Harmonic Analysis最新文献

{"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}

{"title":"Conditional expectation using compactification operators","authors":"Suddhasattwa Das","doi":"10.1016/j.acha.2024.101638","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101638","url":null,"abstract":"<div><p>The separate tasks of denoising, least squares expectation, and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that allow numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems is also shown.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101638"},"PeriodicalIF":2.5,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139732606","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":"Geometric scattering on measure spaces","authors":"Joyce Chew,&nbsp;Matthew Hirn,&nbsp;Smita Krishnaswamy,&nbsp;Deanna Needell,&nbsp;Michael Perlmutter,&nbsp;Holly Steach,&nbsp;Siddharth Viswanath,&nbsp;Hau-Tieng Wu","doi":"10.1016/j.acha.2024.101635","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101635","url":null,"abstract":"<div><p>The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning.</p><p>In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101635"},"PeriodicalIF":2.5,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139710122","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":"Convergent bivariate subdivision scheme with nonnegative mask whose support is non-convex","authors":"Li Cheng","doi":"10.1016/j.acha.2024.101636","DOIUrl":"10.1016/j.acha.2024.101636","url":null,"abstract":"<div><p>Recently we have characterized the convergence of bivariate subdivision scheme with nonnegative mask whose support is convex by means of the so-called connectivity of a square matrix, which is derived by a given mask. The convergence in this case can be checked in linear time with respected to the size of a square matrix. This paper will focus on the characterization of such schemes with non-convex supports.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101636"},"PeriodicalIF":2.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139663000","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":"High-probability generalization bounds for pointwise uniformly stable algorithms","authors":"Jun Fan ,&nbsp;Yunwen Lei","doi":"10.1016/j.acha.2024.101632","DOIUrl":"10.1016/j.acha.2024.101632","url":null,"abstract":"<div><p>Algorithmic stability is a fundamental concept in statistical learning theory to understand the generalization behavior of optimization algorithms. Existing high-probability bounds are developed for the generalization gap as measured by function values and require the algorithm to be uniformly stable. In this paper, we introduce a novel stability measure called pointwise<span> uniform stability by considering the sensitivity of the algorithm with respect to the perturbation of each training example. We show this weaker pointwise uniform stability guarantees almost optimal bounds, and gives the first high-probability bound for the generalization gap as measured by gradients. Sharper bounds<span> are given for strongly convex and smooth problems. We further apply our general result to derive improved generalization bounds for stochastic gradient descent. As a byproduct, we develop concentration inequalities for a summation of weakly-dependent vector-valued random variables.</span></span></p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101632"},"PeriodicalIF":2.5,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139568091","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":"New theoretical insights in the decomposition and time-frequency representation of nonstationary signals: The IMFogram algorithm","authors":"Antonio Cicone ,&nbsp;Wing Suet Li ,&nbsp;Haomin Zhou","doi":"10.1016/j.acha.2024.101634","DOIUrl":"10.1016/j.acha.2024.101634","url":null,"abstract":"<div><p>The analysis of the time–frequency content of a signal is a classical problem in signal processing, with a broad number of applications in real life. Many different approaches have been developed over the decades, which provide alternative time–frequency representations of a signal each with its advantages and limitations. In this work, following the success of nonlinear methods for the decomposition of signals into intrinsic mode functions (IMFs), we first provide more theoretical insights into the so–called Iterative Filtering decomposition algorithm, proving an energy conservation result for the derived decompositions. Furthermore, we present a new time–frequency representation method based on the IMF decomposition of a signal, which is called IMFogram. We prove theoretical results regarding this method, including its convergence to the spectrogram representation for a certain class of signals, and we present a few examples of applications, comparing results with some of the most well-known approaches available in the literature.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101634"},"PeriodicalIF":2.5,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139567875","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":"On representations of the Helmholtz Green's function","authors":"Gregory Beylkin","doi":"10.1016/j.acha.2024.101633","DOIUrl":"10.1016/j.acha.2024.101633","url":null,"abstract":"<div><p>We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number <em>k</em>. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msup><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></mrow></math></span><span><span> operations. The non-oscillatory component has a multiresolution representation via a </span>linear combination of Gaussians and is applied efficiently in space.</span></p><p>Since the Helmholtz Green's function can be viewed as a point source, this partitioning can be interpreted as a splitting into propagating and evanescent components. We show that the non-oscillatory component is significant only in the vicinity of the source at distances <span><math><mi>O</mi><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>⁡</mo><mi>k</mi><mo>)</mo></mrow></math></span>, for some constants <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, whereas the propagating component can be observed at large distances.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101633"},"PeriodicalIF":2.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139544440","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":"Multivariate compactly supported C∞ functions by subdivision","authors":"Maria Charina ,&nbsp;Costanza Conti ,&nbsp;Nira Dyn","doi":"10.1016/j.acha.2024.101630","DOIUrl":"10.1016/j.acha.2024.101630","url":null,"abstract":"<div><p>This paper discusses the generation of multivariate <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called <em>Up-function</em>, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate <em>Up-like functions</em>. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>]</mo></math></span> in comparison to the support <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> compactly supported wavelets of small support in any dimension.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101630"},"PeriodicalIF":2.5,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000071/pdfft?md5=1ad3a9e4a30806ec403f504079a4421d&pid=1-s2.0-S1063520324000071-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505960","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":"Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space","authors":"M. Holler ,&nbsp;A. Schlüter ,&nbsp;B. Wirth","doi":"10.1016/j.acha.2024.101631","DOIUrl":"10.1016/j.acha.2024.101631","url":null,"abstract":"<div><p>An important theme in modern inverse problems is the reconstruction of <em>time-dependent</em> data from only <em>finitely many</em> measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in <em>phase space</em>, the space of positions <em>and</em> velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101631"},"PeriodicalIF":2.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000083/pdfft?md5=ecd67b0e5374d4297b6087dc7c3b9288&pid=1-s2.0-S1063520324000083-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139420410","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}