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Mathematical foundation of sparsitybased multisnapshot spectral estimation Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240607
Ping Liu, Sanghyeon Yu, Ola Sabet, Lucas Pelkmans, Habib AmmariIn this paper, we study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements in a bounded domain. We aim to provide a mathematical foundation for sparsitybased superresolution in such spectral estimation problems in both one and multidimensional spaces. In particular, we estimate the resolution and stability

Adaptive parameter selection for kernel ridge regression Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240607
ShaoBo LinThis paper focuses on parameter selection issues of kernel ridge regression (KRR). Due to special spectral properties of KRR, we find that delicate subdivision of the parameter interval shrinks the difference between two successive KRR estimates. Based on this observation, we develop an earlystopping type parameter selection strategy for KRR according to the socalled Lepskiitype principle. Theoretical

On the existence and estimates of nested spherical designs Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240604
Ruigang Zheng, Xiaosheng Zhuang 
A sharp sufficient condition for mobile sampling in terms of surface density Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240523
Benjamin Jaye, Mishko Mitkovski, Manasa N. Vempati 
Towards a bilipschitz invariant theory Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240514
Jameson Cahill, Joseph W. Iverson, Dustin G. MixonConsider the quotient of a Hilbert space by a subgroup of its automorphisms. We study whether this orbit space can be embedded into a Hilbert space by a bilipschitz map, and we identify constraints on such embeddings.

Gaussian random field approximation via Stein's method with applications to wide random neural networks Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240513
Krishnakumar Balasubramanian, Larry Goldstein, Nathan Ross, Adil SalimWe derive upper bounds on the Wasserstein distance (), with respect to supnorm, between any continuous valued random field indexed by the sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators

Differentially private federated learning with Laplacian smoothing Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240507
Zhicong Liang, Bao Wang, Quanquan Gu, Stanley Osher, Yuan YaoFederated learning aims to protect data privacy by collaboratively learning a model without sharing private data among users. However, an adversary may still be able to infer the private training data by attacking the released model. Differential privacy provides a statistical protection against such attacks at the price of significantly degrading the accuracy or utility of the trained models. In this

The mystery of Carleson frames Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240417
Ole Christensen, Marzieh Hasannasab, Friedrich M. Philipp, Diana Stoeva 
Effectiveness of the tailatomic norm in gridless spectrum estimation Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240416
Wei Li, Shidong Li, Jun Xian 
Complexorder scaleinvariant operators and selfsimilar processes Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240404
Arash Amini, Julien Fageot, Michael UnserIn this paper, we perform the joint study of scaleinvariant operators and selfsimilar processes of complex order. More precisely, we introduce general families of scaleinvariant complexorder fractionalderivation and integration operators by constructing them in the Fourier domain. We analyze these operators in detail, with special emphasis on the decay properties of their output. We further use

Error bounds for kernelbased approximations of the Koopman operator Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240404
Friedrich M. Philipp, Manuel Schaller, Karl Worthmann, Sebastian Peitz, Feliks NüskeWe consider the datadriven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from longterm ergodic simulations. We derive both an exact expression for the variance of the kernel crosscovariance operator, measured in the HilbertSchmidt norm, and probabilistic bounds

Frame set for Gabor systems with Haar window Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240318
XinRong Dai, Meng ZhuWe describe the full structure of the frame set for the Gabor system with the window being the Haar function . This is the first compactly supported window function for which the frame set is represented explicitly.

Frame set for shifted sincfunction Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240316
Yurii Belov, Andrei V. Semenov 
Eigenmatrix for unstructured sparse recovery Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240314
Lexing Ying 
Solving PDEs on unknown manifolds with machine learning Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240229
Senwei Liang, Shixiao W. Jiang, John Harlim, Haizhao YangThis paper proposes a meshfree 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 leastsquares 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 qsphere Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240229
Frank Filbir, Ralf Hielscher, Thomas Jahn, Tino UllrichThe 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

Separationfree spectral superresolution via convex optimization Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240229
Zai Yang, YiLin Mo, Zongben XuAtomic norm methods have recently been proposed for spectral superresolution 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 signaltonoise (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 timefrequency localization operator before the plunge region Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240228
Aleksei KulikovFor a pair of sets the timefrequency 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 2.6) Pub Date : 20240227
Alex Barnett, Philip Greengard, Manas RachhThe 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 2.6) Pub Date : 20240222
Vlado Menkovski, Jacobus W. Portegies, Mahefa Ratsisetraina RavelonanosyWe 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 Ginvariant graph Laplacian Part I: Convergence rate and eigendecomposition Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240221
Eitan Rosen, Paulina Hoyos, Xiuyuan Cheng, Joe Kileel, Yoel ShkolniskyGraph 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 . We propose to construct the graph Laplacian by incorporating the distances between all the pairs of

Variable bandwidth via Wilson bases Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240221
Beatrice Andreolli, Karlheinz GröchenigWe 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 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


Conditional expectation using compactification operators Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240209
Suddhasattwa DasThe 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 2.6) Pub Date : 20240206
Joyce Chew, Matthew Hirn, Smita Krishnaswamy, Deanna Needell, Michael Perlmutter, Holly Steach, Siddharth Viswanath, HauTieng WuThe scattering transform is a multilayered, waveletbased 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 nonEuclidean structure, such

Convergent bivariate subdivision scheme with nonnegative mask whose support is nonconvex Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240201
Li ChengRecently we have characterized the convergence of bivariate subdivision scheme with nonnegative mask whose support is convex by means of the socalled 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 nonconvex

Highprobability generalization bounds for pointwise uniformly stable algorithms Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240127
Jun Fan, Yunwen LeiAlgorithmic stability is a fundamental concept in statistical learning theory to understand the generalization behavior of optimization algorithms. Existing highprobability 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

New theoretical insights in the decomposition and timefrequency representation of nonstationary signals: The IMFogram algorithm Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240126
Antonio Cicone, Wing Suet Li, Haomin ZhouThe 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

On representations of the Helmholtz Green's function Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240124
Gregory BeylkinWe consider the free space Helmholtz Green's function and split it into the sum of oscillatory and nonoscillatory (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 2.6) Pub Date : 20240119
Maria Charina, Costanza Conti, Nira DynThis paper discusses the generation of multivariate C∞ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Upfunction, by a nonstationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Uplike functions. We generate them by nonstationary schemes based

Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240111
M. Holler, A. Schlüter, B. WirthAn important theme in modern inverse problems is the reconstruction of timedependent 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 divideandconquer algorithm for distributed optimization on networks Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20240102
Nazar Emirov, Guohui Song, Qiyu SunIn 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 peertopeer 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 divideandconquer algorithm to solve


Timefrequency analysis on flat tori and Gabor frames in finite dimensions Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20231212
L.D. Abreu, P. Balazs, N. Holighaus, F. Luef, M. SpeckbacherWe provide the foundations of a Hilbert space theory for the shorttime Fourier transform (STFT) where the flat tori TN2=R2/(Z×NZ)=[0,1]×[0,N] act as phase spaces. We work on an Ndimensional 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 spatiospectral limiting operators in higher dimensions Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20231213
Arie Israel, Azita MayeliProlate 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 timefrequency 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

LaplaceBeltrami operator on the orthogonal group in ambient (Euclidean) coordinates Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20231212
Petre Birtea, Ioan Caşu, Dan ComănescuUsing the embedded gradient vector field method (see P. Birtea, D. Comănescu (2015) [7]), we present a general formula for the LaplaceBeltrami 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 LaplaceBeltrami operator on the orthogonal

Spline manipulations for empirical mode decomposition (EMD) on bounded intervals and beyond Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20231205
Charles K. Chui, Wenjie HeEmpirical mode decomposition (EMD), introduced by N.E. Huang et al. in 1998, is perhaps the most popular datadriven computational scheme for the decomposition of a nonstationary signal or time series f(t), with timedomain 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 multipenalty distribution regression Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20231123
Zhan Yu, Daniel W.C. HoThis paper is concerned with functional learning by utilizing twostage sampled distribution regression. We study a multipenalty regularization algorithm for distribution regression in the framework of learning theory. The algorithm aims at regressing to realvalued 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 2.6) Pub Date : 20231117
Hartmut Führ, Reihaneh RaisiTousiWe investigate the invariance properties of general wavelet coorbit spaces and Besovtype decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasiisometry 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 2.6) Pub Date : 20231108
Hartmut Führ, Irina ShafkulovskaWe 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) welldefined, (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 2.6) Pub Date : 20231110
Amine Laghrib, Lekbir AfraitesImage denoising is always considered an important area of image processing. In this work, we address a new PDEbased 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 2.6) Pub Date : 20231108
Jinjun Li, Zhiyi WuWe 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 2.6) Pub Date : 20231029
Götz Pfander, Shauna Revay, David WalnutFor 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∈JIj. The construction extends to infinite

A multivariate Riesz basis of ReLU neural networks Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20231020
Cornelia Schneider, Jan VybíralWe consider the trigonometriclike 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 2.6) Pub Date : 20231013
Anton Kutsenko, Sergey Danilov, Stephan Juricke, Marcel OliverWe 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 2.6) Pub Date : 20231012
Hao Liu, Alex Havrilla, Rongjie Lai, Wenjing LiaoAutoencoders have demonstrated remarkable success in learning lowdimensional latent features of highdimensional data across various applications. Assuming that data are sampled near a lowdimensional manifold, we employ chart autoencoders, which encode data into lowdimensional 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 2.6) Pub Date : 20231010
M.M. Castro, F.A. Grünbaum, I. ZurriánThe “timeandband 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 2.6) Pub Date : 20231006
Yucong Liu, Simiao Jiao, LekHeng LimAny matrix A has an LU decomposition up to a row or column permutation. Less wellknown 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 2.6) Pub Date : 20231005
Peter Balazs, Mitra Shamsabadi, Ali Akbar Arefijamaal, Gilles ChardonTo 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 2.6) Pub Date : 20230909
Albert Chua, Matthew Hirn, Anna LittleIn 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 welldefined, 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 2.6) Pub Date : 20230909
Ryan Vaughn, Tyrus Berry, Harbir AntilGiven 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 timefrequency spaces Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20230909
Elena Cordero, Gianluca GiacchiWe 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 fullgeneral 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 2.6) Pub Date : 20230906
HungHsu Chou, Carsten Gieshoff, Johannes Maly, Holger RauhutIn deep learning, it is common to use more network parameters than training points. In such scenario of overparameterization, 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 2.6) Pub Date : 20230821
Tal Shnitzer, HauTieng Wu, Ronen TalmonMultivariate timeseries have become abundant in recent years, as many dataacquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operatorbased approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators

Performance bounds of the intensitybased estimators for noisy phase retrieval Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20230819
Meng Huang, Zhiqiang XuThe 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 intensitybased loss function, i.e., xˆ:=argminx∈Cd∑j=1m(〈aj,x〉2−bj)2. Although many algorithms for solving the intensitybased estimator have

Learning ability of interpolating deep convolutional neural networks Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20230816
TianYi Zhou, Xiaoming HuoIt is frequently observed that overparameterized neural networks generalize well. Regarding such phenomena, existing theoretical work mainly devotes to linear settings or fullyconnected 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 2.6) Pub Date : 20230804
Neda Mohammadi, Victor M. PanaretosIs 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 piecewiselinear functions Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20230809
Alexis Goujon, Joaquim Campos, Michael UnserRectifiedlinearunit (ReLU) neural networks, which play a prominent role in deep learning, generate continuous and piecewiselinear (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 2.6) Pub Date : 20230802
Joachim Toft, Divyang G. Bhimani, Ramesh MannaWe 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 temporalattention product Appl. Comput. Harmon. Anal. (IF 2.6) Pub Date : 20230728
Ru Geng, Yixian Gao, HongKun Zhang, Jian ZuSignal 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 temporalattention 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' spatiotemporal