样式: 排序: IF: - GO 导出 标记为已读
-
Solving PDEs on unknown manifolds with machine learning Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-29 Senwei Liang, Shixiao W. Jiang, John Harlim, Haizhao Yang
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)
-
Marcinkiewicz–Zygmund inequalities for scattered and random data on the q-sphere Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-29 Frank Filbir, Ralf Hielscher, Thomas Jahn, Tino Ullrich
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 -dimensional sphere , and investigate how well continuous -norms of polynomials of maximum degree on the sphere
-
Separation-Free Spectral Super-Resolution via Convex Optimization Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-29 Zai Yang, Yi-Lin Mo, Gongguo Tang, Zongben Xu
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
-
Exponential lower bound for the eigenvalues of the time-frequency localization operator before the plunge region Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-28 Aleksei Kulikov
For a pair of sets the time-frequency localization operator is defined as , where is the Fourier transform and are projection operators onto and Ω, respectively. We show that in the case when both and Ω are intervals, the eigenvalues of satisfy if , where is arbitrary and , provided that . This improves the result of Bonami, Jaming and Karoui, who proved it for . The proof is based on the properties
-
Uniform approximation of common Gaussian process kernels using equispaced Fourier grids Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-27 Alex Barnett, Philip Greengard, Manas Rachh
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
-
Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-22 Vlado Menkovski, Jacobus W. Portegies, Mahefa Ratsisetraina Ravelonanosy
We give an asymptotic expansion of the relative entropy between the heat kernel of a compact Riemannian manifold and the normalized Riemannian volume for small values of and for a fixed element . 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 , when they are expressed in terms of normal
-
The G-invariant graph Laplacian Part I: Convergence rate and eigendecomposition Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-21 Eitan Rosen, Paulina Hoyos, Xiuyuan Cheng, Joe Kileel, Yoel Shkolnisky
-
Variable bandwidth via Wilson bases Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-21 Beatrice Andreolli, Karlheinz Gröchenig
-
-
Conditional expectation using compactification operators Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-09 Suddhasattwa Das
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
-
Geometric scattering on measure spaces Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-06 Joyce Chew, Matthew Hirn, Smita Krishnaswamy, Deanna Needell, Michael Perlmutter, Holly Steach, Siddharth Viswanath, Hau-Tieng Wu
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
-
Convergent bivariate subdivision scheme with nonnegative mask whose support is non-convex Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-02-01 Li Cheng
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
-
New theoretical insights in the decomposition and time-frequency representation of nonstationary signals: the IMFogram algorithm Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-01-26 Antonio Cicone, Wing Suet Li, Haomin Zhou
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
-
High-probability generalization bounds for pointwise uniformly stable algorithms Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-01-27 Jun Fan, Yunwen Lei
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 uniform stability by considering
-
On representations of the Helmholtz Green's function Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-01-24 Gregory Beylkin
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 k. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be
-
Multivariate compactly supported C∞ functions by subdivision Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-01-19 Maria Charina, Costanza Conti, Nira Dyn
This paper discusses the generation of multivariate C∞ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based
-
Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-01-11 M. Holler, A. Schlüter, B. Wirth
An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many 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 phase space, the space of positions
-
A divide-and-conquer algorithm for distributed optimization on networks Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2024-01-02 Nazar Emirov, Guohui Song, Qiyu Sun
In this paper, we consider networks with topologies described by some connected undirected graph G=(V,E) and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem minx{F(x)=∑i∈Vfi(x)} with local objective functions fi depending only on neighboring variables of the vertex i∈V. We introduce a divide-and-conquer algorithm to solve
-
-
Time-frequency analysis on flat tori and Gabor frames in finite dimensions Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-12-12 L.D. Abreu, P. Balazs, N. Holighaus, F. Luef, M. Speckbacher
We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori TN2=R2/(Z×NZ)=[0,1]×[0,N] act as phase spaces. We work on an N-dimensional subspace SN of distributions periodic in time and frequency in the dual S0′(R) of the Feichtinger algebra S0(R) and equip it with an inner product. To construct the Hilbert space SN we apply a suitable double
-
On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-12-13 Arie Israel, Azita Mayeli
Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO
-
Laplace-Beltrami operator on the orthogonal group in ambient (Euclidean) coordinates Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-12-12 Petre Birtea, Ioan Caşu, Dan Comănescu
Using the embedded gradient vector field method (see P. Birtea, D. Comănescu (2015) [7]), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of n×n matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal
-
Spline manipulations for empirical mode decomposition (EMD) on bounded intervals and beyond Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-12-05 Charles K. Chui, Wenjie He
Empirical mode decomposition (EMD), introduced by N.E. Huang et al. in 1998, is perhaps the most popular data-driven computational scheme for the decomposition of a non-stationary signal or time series f(t), with time-domain R:=(−∞,∞), into finitely many oscillatory components {f1(t),⋯,fK(t)}, called intrinsic mode functions (IMFs), and some “almost monotone” remainder r(t), called the trend of f(t)
-
Estimates on learning rates for multi-penalty distribution regression Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-11-23 Zhan Yu, Daniel W.C. Ho
This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression in the framework of learning theory. The algorithm aims at regressing to real-valued outputs from probability measures. The theoretical analysis of distribution regression is far from maturity and quite challenging since
-
Dilational symmetries of decomposition and coorbit spaces Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-11-17 Hartmut Führ, Reihaneh Raisi-Tousi
We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings. We then apply the general results
-
The metaplectic action on modulation spaces Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-11-08 Hartmut Führ, Irina Shafkulovska
We study the mapping properties of metaplectic operators Sˆ∈Mp(2d,R) on modulation spaces of the type Mmp,q(Rd). Our main result is a full characterization of the pairs (Sˆ,Mp,q(Rd)) for which the operator Sˆ:Mp,q(Rd)→Mp,q(Rd) is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that Sˆ is a Banach space automorphism. For polynomially bounded weight
-
Image denoising based on a variable spatially exponent PDE Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-11-10 Amine Laghrib, Lekbir Afraites
Image denoising is always considered an important area of image processing. In this work, we address a new PDE-based model for image denoising that have been contaminated by multiplicative noise, specially the Speckle one. We propose a new class of PDEs whose nonlinear structure depends on a spatially tensor depending quantity attached to the desired solution, which takes into account the gray level
-
On the intermediate value property of spectra for a class of Moran spectral measures Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-11-08 Jinjun Li, Zhiyi Wu
We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are in 0 and their upper entropy dimensions. Moreover, for such a Moran spectral measure μ, we show that the Beurling dimension for the spectra of μ has the intermediate value property: let t be any value in 0 and the upper entropy dimension of μ, then there exists a spectrum whose Beurling dimension is t. In
-
Exponential bases for partitions of intervals Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-29 Götz Pfander, Shauna Revay, David Walnut
For a partition of [0,1] into intervals I1,…,In we prove the existence of a partition of Z into Λ1,…,Λn such that the complex exponential functions with frequencies in Λk form a Riesz basis for L2(Ik), and furthermore, that for any J⊆{1,2,…,n}, the exponential functions with frequencies in ⋃j∈JΛj form a Riesz basis for L2(I) for any interval I with length |I|=∑j∈J|Ij|. The construction extends to infinite
-
A multivariate Riesz basis of ReLU neural networks Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-20 Cornelia Schneider, Jan Vybíral
We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of L2([0,1]) based on the Gershgorin theorem. We also generalize this system to higher dimensions d>1 by a construction, which avoids using (tensor) products. As a consequence, the functions
-
On the relation between Fourier and Walsh–Rademacher spectra for random fields Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-13 Anton Kutsenko, Sergey Danilov, Stephan Juricke, Marcel Oliver
We discuss relations between the expansion coefficients of a discrete random field when analyzed with respect to different hierarchical bases. Our main focus is on the comparison of two such systems: the Walsh–Rademacher basis and the trigonometric Fourier basis. In general, spectra computed with respect to one basis will look different in the other. In this paper, we prove that, in a statistical sense
-
Deep nonparametric estimation of intrinsic data structures by chart autoencoders: Generalization error and robustness Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-12 Hao Liu, Alex Havrilla, Rongjie Lai, Wenjing Liao
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
-
Time and band limiting for exceptional polynomials Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-10 M.M. Castro, F.A. Grünbaum, I. Zurrián
The “time-and-band limiting” commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones. Here we give a general result that insures
-
LU decomposition and Toeplitz decomposition of a neural network Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-06 Yucong Liu, Simiao Jiao, Lek-Heng Lim
Any matrix A has an LU decomposition up to a row or column permutation. Less well-known is the fact that it has a ‘Toeplitz decomposition’ A=T1T2⋯Tr where Ti's are Toeplitz matrices. We will prove that any continuous function f:Rn→Rm has an approximation to arbitrary accuracy by a neural network that maps x∈Rn to L1σ1U1σ2L2σ3U2⋯Lrσ2r−1Urx∈Rm, i.e., where the weight matrices alternate between lower
-
Representation of operators using fusion frames Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-10-05 Peter Balazs, Mitra Shamsabadi, Ali Akbar Arefijamaal, Gilles Chardon
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
-
On generalizations of the nonwindowed scattering transform Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-09-09 Albert Chua, Matthew Hirn, Anna Little
In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as Lq(Rn) norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of C2 diffeomorphisms in specific cases. Lastly, we extend
-
Diffusion maps for embedded manifolds with boundary with applications to PDEs Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-09-09 Ryan Vaughn, Tyrus Berry, Harbir Antil
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
-
Metaplectic Gabor frames and symplectic analysis of time-frequency spaces Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-09-09 Elena Cordero, Gianluca Giacchi
We introduce new frames, called metaplectic Gabor frames, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions, cf. [7], [8], [5], [17], [27], [28]. Namely, we develop the theory of metaplectic atoms in a full-general setting and prove an inversion formula for metaplectic Wigner distributions on Rd. Its discretization provides metaplectic Gabor frames. Next
-
Gradient descent for deep matrix factorization: Dynamics and implicit bias towards low rank Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-09-06 Hung-Hsu Chou, Carsten Gieshoff, Johannes Maly, Holger Rauhut
In deep learning, it is common to use more network parameters than training points. In such scenario of over-parameterization, there are usually multiple networks that achieve zero training error so that the training algorithm induces an implicit bias on the computed solution. In practice, (stochastic) gradient descent tends to prefer solutions which generalize well, which provides a possible explanation
-
Spatiotemporal analysis using Riemannian composition of diffusion operators Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-08-21 Tal Shnitzer, Hau-Tieng Wu, Ronen Talmon
Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators
-
Performance bounds of the intensity-based estimators for noisy phase retrieval Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-08-19 Meng Huang, Zhiqiang Xu
The aim of noisy phase retrieval is to estimate a signal x0∈Cd from m noisy intensity measurements bj=|〈aj,x0〉|2+ηj,j=1,…,m, where aj∈Cd are known measurement vectors and η=(η1,…,ηm)⊤∈Rm is a noise vector. A commonly used estimator for x0 is to minimize the intensity-based loss function, i.e., xˆ:=argminx∈Cd∑j=1m(|〈aj,x〉|2−bj)2. Although many algorithms for solving the intensity-based estimator have
-
Learning ability of interpolating deep convolutional neural networks Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-08-16 Tian-Yi Zhou, Xiaoming Huo
It is frequently observed that overparameterized neural networks generalize well. Regarding such phenomena, existing theoretical work mainly devotes to linear settings or fully-connected neural networks. This paper studies the learning ability of an important family of deep neural networks, deep convolutional neural networks (DCNNs), under both underparameterized and overparameterized settings. We
-
Detecting whether a stochastic process is finitely expressed in a basis Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-08-04 Neda Mohammadi, Victor M. Panaretos
Is it possible to detect if the sample paths of a stochastic process almost surely admit a finite expansion with respect to some/any basis? The determination is to be made on the basis of a finite collection of discretely/noisily observed sample paths. We show that it is indeed possible to construct a hypothesis testing scheme that is almost surely guaranteed to make only finite many incorrect decisions
-
Stable parameterization of continuous and piecewise-linear functions Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-08-09 Alexis Goujon, Joaquim Campos, Michael Unser
Rectified-linear-unit (ReLU) neural networks, which play a prominent role in deep learning, generate continuous and piecewise-linear (CPWL) functions. While they provide a powerful parametric representation, the mapping between the parameter and function spaces lacks stability. In this paper, we investigate an alternative representation of CPWL functions that relies on local hat basis functions and
-
Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipović and modulation spaces Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-08-02 Joachim Toft, Divyang G. Bhimani, Ramesh Manna
We give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates for such operators on modulation spaces, and apply the results to prove Strichartz estimates for such propagators when acting on Pilipović and modulation spaces. Especially we extend some results by Balhara, Cordero, Nicola, Rodino
-
Graph signal processing on dynamic graphs based on temporal-attention product Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-07-28 Ru Geng, Yixian Gao, Hong-Kun Zhang, Jian Zu
Signal processing is an important research topic. This paper aims to provide a general framework for signal processing on arbitrary dynamic graphs. We propose a new graph transformation by defining a temporal-attention product. This product transforms the sequence of graph time slices with arbitrary topology and number of nodes into a static graph, effectively capturing graph signals' spatio-temporal
-
A note on spike localization for line spectrum estimation Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-07-27 Haoya Li, Hongkang Ni, Lexing Ying
This note considers the problem of approximating the locations of dominant spikes for a probability measure from noisy spectrum measurements under the condition of residue signal, significant noise level, and no minimum spectrum separation. We show that the simple procedure of thresholding the smoothed inverse Fourier transform allows for approximating the spike locations rather accurately.
-
Random sampling over locally compact Abelian groups and inversion of the Radon transform Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-07-25 Erika Porten, Juan Miguel Medina, Marcela Morvidone
We consider the problem of reconstructing a measurable function over a Locally Compact Abelian group G from random measurements. The results presented herein are partially inspired by the concept of alias-free sampling. Here, the sampling and interpolation operation is modelled as an approximate convolution operator with respect to a stochastic integral defined with an appropriately chosen random measure
-
Generative modeling via tensor train sketching Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-07-17 YoonHaeng Hur, Jeremy G. Hoskins, Michael Lindsey, E.M. Stoudenmire, Yuehaw Khoo
In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that
-
Gabor frame bound optimizations Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-07-13 Markus Faulhuber, Irina Shafkulovska
We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the frame bounds and the condition number are rectangular lattices which are different for the respective quantities. Also, in some cases optimal lattices do not exist
-
Corrigendum to “A diffusion + wavelet-window method for recovery of super-resolution point-masses with application to single-molecule microscopy and beyond” [Appl. Comput. Harmon. Anal. 63 (2023) 1–19] Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-30 Charles K. Chui
This note is to point out a serious typo in [1] and clarify a notation in [2].
-
-
A constructive approach for computing the proximity operator of the p-th power of the ℓ1 norm Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-29 Ashley Prater-Bennette, Lixin Shen, Erin E. Tripp
This note is to study the proximity operator of hp=‖⋅‖1p, the power function of the ℓ1 norm. For general p, computing the proximity operator requires solving a system of potentially highly nonlinear inclusions. For p=1, the proximity operator of h1 is the well known soft-thresholding operator. For p=2, the function h2 serves as a penalty function that promotes structured solutions to optimization problems
-
Analytic and directional wavelet packets in the space of periodic signals Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-26 Amir Averbuch, Pekka Neittaanmäki, Valery Zheludev
The paper presents a versatile library of analytic and quasi-analytic complex-valued wavelet packets (WPs) which originate from discrete splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based orthonormal WPs designed in [4]. The imaginary parts are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, are antisymmetric. Tensor
-
n-Best kernel approximation in reproducing kernel Hilbert spaces Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-22 Tao Qian
By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy
-
Robust Sensing of Low-Rank Matrices with Non-Orthogonal Sparse Decomposition Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-20 Johannes Maly
We consider the problem of recovering an unknown low-rank matrix X⋆ with (possibly) non-orthogonal, effectively sparse rank-1 decomposition from measurements y gathered in a linear measurement process A. We propose a variational formulation that lends itself to alternating minimization and whose global minimizers provably approximate X⋆ up to noise level. Working with a variant of robust injectivity
-
Capacity dependent analysis for functional online learning algorithms Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-19 Xin Guo, Zheng-Chu Guo, Lei Shi
This article provides convergence analysis of online stochastic gradient descent algorithms for functional linear models. Adopting the characterizations of the slope function regularity, the kernel space capacity, and the capacity of the sampling process covariance operator, significant improvement on the convergence rates is achieved. Both prediction problems and estimation problems are studied, where
-
Decentralized learning over a network with Nyström approximation using SGD Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-16 Heng Lian, Jiamin Liu
Nowadays we often meet with a learning problem when data are distributed on different machines connected via a network, instead of stored centrally. Here we consider decentralized supervised learning in a reproducing kernel Hilbert space. We note that standard gradient descent in a reproducing kernel Hilbert space is difficult to implement with multiple communications between worker machines. On the
-
A unified approach to synchronization problems over subgroups of the orthogonal group Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-14 Huikang Liu, Man-Chung Yue, Anthony Man-Cho So
The problem of synchronization over a group G aims to estimate a collection of group elements G1⁎,…,Gn⁎∈G based on noisy observations of a subset of all pairwise ratios of the form Gi⁎Gj⁎−1. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the
-
Estimation under group actions: Recovering orbits from invariants Appl. Comput. Harmon. Anal. (IF 2.5) Pub Date : 2023-06-08 Afonso S. Bandeira, Ben Blum-Smith, Joe Kileel, Jonathan Niles-Weed, Amelia Perry, Alexander S. Wein
We study a class of orbit recovery problems in which we observe independent copies of an unknown element of Rp, each linearly acted upon by a random element of some group (such as Z/p or SO(3)) and then corrupted by additive Gaussian noise. We prove matching upper and lower bounds on the number of samples required to approximately recover the group orbit of this unknown element with high probability