We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. Thus the diffusion

We develop a new harmonic analysis of reproducing kernels. Our approach is with view to a number of recent applications: We explore both geometric and algorithmic consequences of our analysis. The list of applications includes: feature fitting in machine learning algorithms, dynamics in large networks, spline approximation, determinantal point processes, stochastic processes, and fractals.

We study the problem of reconstructing a positive discrete measure on a compact set K⊆Rn from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.

We use the well-known observation that the solutions of Jacobi's differential equation can be represented via the non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial transforms. More explicitly, it follows from this observation that the matrix corresponding to the discrete Jacobi transform is the Hadamard product of a numerically

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that compressible signals admit when sampled along spirals

Approximation properties of multivariate quasi-projection operators are studied. Wide classes of such operators are considered, including the sampling and the Kantorovich-Kotelnikov type operators generated by different band-limited functions. The rate of convergence in the weighted Lp-spaces for these operators is investigated. The results allow us to estimate the error for reconstruction of signals

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the ℓ1-analysis basis pursuit, enabling accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend

A sequence of continuous real functions (fm)m∈N0, with exponential decay at infinity that implies their Fourier-Laplace transforms Fm(iz) are analytic on |ℜ(z)|

Our goal in the paper is to address the following problem: From an unknown matrix, we are given inner products of that matrix with a set of prescribed matrices, and wish to find the unknown matrix. We shall consider this problem by using the notion of minimal norm interpolation. A fixed-point proximity algorithm for solving this problem will be developed.

In this paper, we study the spectral property of a class of self-affine measures μR,D on R2 generated by the iterated function system {ϕd(⋅)=R−1(⋅+d)}d∈D associated with the real expanding matrix R=(b100b2) and the digit set D={(00),(10),(01)}. We show that μR,D is a spectral measure if and only if 3|bi, i=1,2. This extends the result of Deng and Lau [J. Funct. Anal., 2015], where they considered the

We consider the class of Rudin–Shapiro-like polynomials, whose L4 norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f(z)=f0+f1z+⋯+fdzd is identified with the sequence (f0,f1,…,fd) of its coefficients. From the L4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the

In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order α>−1 on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both an integral operator, and a Sturm–Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalization of orthogonal ball polynomials in primitive

Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments

This paper studies sensor calibration in spectral estimation where the true frequencies are located on a continuous domain. We consider a uniform array of sensors that collects measurements whose spectrum is composed of a finite number of frequencies, where each sensor has an unknown calibration parameter. Our goal is to recover the spectrum and the calibration parameters simultaneously from multiple

This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard matrices) and wavelet reconstruction the stable sampling rate is linear. This implies that binary measurements are as efficient as Fourier samples when using wavelets as

Last decade saw the creation of a number of directional representation dictionaries that desire to address the weaknesses of the classical wavelet transform that arise due to its limited capacity for the analysis of edge-like features of two-dimensional signals. Salient features of these dictionaries are directional selectivity and anisotropic treatment of the axes, achieved through the parabolic scaling

In a recent paper, Flandrin [16] proposed filtering based on the zeros of a spectrogram with Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time–frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram

Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which

It was recently shown that on a large class of important Banach spaces there exist no linear methods which are able to approximate the Hilbert transform from samples of the given function. This implies that there is no linear algorithm for calculating the Hilbert transform which can be implemented on a digital computer and which converges for all functions from the corresponding Banach spaces. The

In this article, we investigate the privacy issues that arise from a new frame-based kernel analysis approach to reconstruct from frame coefficient erasures. We show that while an erasure recovery matrix is needed in addition to a decoding frame for a receiver to recover the erasures, the erasure recovery matrix can be designed in such a way that it protects the encoding frame. The set of such erasure

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. Deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the approximation or generalization ability of deep learning methods generated by the network architectures such

In recent years, correntropy and its applications in machine learning have been drawing continuous attention owing to its merits in dealing with non-Gaussian noise and outliers. However, theoretical understanding of correntropy, especially in the learning theory context, is still limited. In this study, we investigate correntropy based regression in the presence of non-Gaussian noise or outliers within

This paper proposes the multiresolution mode decomposition (MMD) as a novel model for adaptive time series analysis. The main conceptual innovation is the introduction of the multiresolution intrinsic mode function (MIMF) of the form∑n=−N/2N/2−1ancos⁡(2πnϕ(t))scn(2πNϕ(t))+∑n=−N/2N/2−1bnsin⁡(2πnϕ(t))ssn(2πNϕ(t)) to model nonlinear and non-stationary data with time-dependent amplitudes, frequencies,

Strictly proper kernel scores are well-known tools in probabilistic forecasting, while characteristic kernels have been extensively investigated in machine learning. We show that both notions coincide, so insights from one part of the literature can be used in the other. We show that the metric induced by a characteristic kernel cannot reliably distinguish between distributions that are far apart in

The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-à-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a new model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays

In fast direct solvers for integral equations with the Laplace kernel, a hierarchical compression process needs to compute interpolative decompositions of off-diagonal block rows of the discretized integral operators. This computation can be dramatically accelerated by a technique called the proxy surface method, which is motivated by potential theory. We present a long overdue, rigorous error analysis

Dimensionality reduction is in demand to reduce the complexity of solving large-scale problems with data lying in latent low-dimensional structures in machine learning and computer vision. Motivated by such need, in this work we study the Restricted Isometry Property (RIP) of Gaussian random projections for low-dimensional subspaces in RN, and rigorously prove that the projection Frobenius norm distance

We consider a causal FIR symmetric paraunitary(PU) matrix extension with parameters, and the constructions of symmetric orthogonal wavelets and symmetric tight framelets with an integer dilation factor M⩾2. Firstly, we propose an algorithm for factorizing a causal FIR symmetric Laurent polynomial vector into the product of order-one paraunitary matrices and constant vector. Secondly, based on the factorization

We present a numerical algorithm for an accurate and efficient computation of the convolution of the frequency domain layered media Green's function with a given density function. Instead of compressing the convolution matrix directly as in the classical fast multipole, fast direct solver, and H-matrix algorithms, this new algorithm considers a translated form of the original matrix so existing blocks

The uncertainty product of a function is a quantity that measures the trade-off between the space and the frequency localization of the function. Its boundedness from below is the content of various uncertainty principles. In the present paper, functions over the n-dimensional sphere are considered. A formula is derived that expresses the uncertainty product of a continuous function in terms of its

Knowing that the convergence of a multivariate subdivision scheme with a nonnegative mask can be characterized by whether or not some finite products of row-stochastic matrices induced by this mask have a positive column. However, the number of those products is exponential with respect to the size of matrices. For nonnegative univariate subdivision, this problem is completely solved. Thus, the convergence

The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the other hand, in finite-dimensional settings, unique solvability of the problem implies uniform stability. A prominent example of such a phase retrieval

The aim of matrix recovery is to recover P∈M⊂Fp×q from LA(P)=(Tr(A1TP),Tr(A2TP),…,Tr(ANTP))T with Aj∈Vj⊂Fp×q, which is raised in many areas. In this paper, we build up a framework for almost everywhere matrix recovery which means LA is almost everywhere injectivity on M. We mainly focus on the following question: how many measurements are needed to recover almost all the matrices in M? For the case

In this work we give a sense to the notion of orientation for self-similar Gaussian fields with stationary increments, based on a Riesz analysis of these fields, with isotropic zero-mean analysis functions. We propose a structure tensor formulation and provide an intrinsic definition of the orientation vector as eigenvector of this tensor. That is, we show that the orientation vector does not depend

We prove sharp upper and lower bounds for generalized Calderón's sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations and general measurable families of automorphisms. The proof makes use of techniques of analysis on metric spaces, and relies on a counting estimate of lattice points inside metric balls. We will deduce as special cases Calderón-type

In this paper, we investigate the active set identification technique of ISTA and provide some good properties. An active set Newton-CG method is then proposed for ℓ1 optimization. Under appropriate conditions, we show that the proposed method is globally convergent with some nonmonotone line search. The numerical comparisons with several state-of-art methods demonstrate the efficiency of the proposed

This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization of CUR decompositions which synergizes these viewpoints and shows that they are indeed the same in the exact decomposition case.

Kernel-based nonlinear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph associated with the data set. When each individual data object carries additional structural details, however, the correspondence relations between these structures provide

We consider the problem of non-negative super-resolution, which concerns reconstructing a non-negative signal x=∑i=1kaiδti from m samples of its convolution with a window function ϕ(s−t), of the form y(sj)=∑i=1kaiϕ(sj−ti)+δj, where δj indicates an inexactness in the sample value. We first show that x is the unique non-negative measure consistent with the samples, provided the samples are exact. Moreover

An equiangular tight frame (ETF) is a type of optimal packing of lines. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have unimodular entries. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they

We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As

A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of Gaussian white noises [4]. This answered pioneering work by Flandrin [10], who observed that the zeros of the Gabor transform of white noise had a regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. In this paper, we study in a systematic way the link between GAFs

There are two major routes to address inverse problems in signal and image processing, such as denoising or deblurring. A first route relies on Bayesian modeling, where prior probabilities embody models of both the distribution of the unknown variables and their statistical dependence with respect to the observed data. Estimation typically relies on the minimization of an expected loss (e.g. minimum

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type Ln−l−1(l+1/2)(r2)rlYlm(ϑ,φ), |m|≤l

The continuous wavelet transform (CWT)-based synchrosqueezing transform (SST) is a special type of the reassignment method which not only enhances the energy concentration of CWT in the time-frequency plane, but also separates the components of multicomponent signals. The “bump wavelet” and Morlet's wavelet are commonly used continuous wavelets for SST. There is a parameter in these wavelets which

We generalize the scattering transform to graphs and consequently construct a convolutional neural network on graphs. We show that under certain conditions, any feature generated by such a network is approximately invariant to permutations and stable to signal and graph manipulations. Numerical results demonstrate competitive performance on relevant datasets.

In this paper, we consider the design of wavelet filters based on the Thiran's common-factor approach proposed in [13]. This approach aims at building finite impulse response filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complex wavelet. Unfortunately it is not possible to construct wavelets which are both finitely supported and analytic. The wavelet filters constructed

Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on multireference alignment. The recent non-unique games framework provides a representation theoretic

In this paper, we consider the phase retrieval problem for recovering a complex signal, given a number of observations on the magnitude of linear measurements. This problem has direct applications in X-ray crystallography, diffraction imaging and microscopy. Motivated by the extensively studied theory of (tight) wavelet frame and its great success in various applications, we propose a wavelet frame

In Group Synchronization, one attempts to recover a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization

The robustness of manifold learning methods is often predicated on the stability of the Neumann Laplacian eigenfunctions under deformations of the assumed underlying domain. Indeed, many manifold learning methods are based on approximating the Neumann Laplacian eigenfunctions on a manifold that is assumed to underlie data, which is viewed through a source of distortion. In this paper, we study the

We introduce the prime coset sum method for constructing tight wavelet frames, which allows one to construct nonseparable multivariate tight wavelet frames with prime dilation, using a univariate lowpass mask with this same prime dilation as input. This method relies on the idea of finding a sum of hermitian squares representation for a nonnegative trigonometric polynomial related to the sub-QMF condition

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices A∈GL(d,R), with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial

In this paper, we consider the recovery of group sparse signals corrupted by impulsive noise. In some recent literature, researchers have utilized stable data fitting models, like l1-norm, Huber penalty function and Lorentzian-norm, to substitute the l2-norm data fidelity model to obtain more robust performance. In this paper, a stable model is developed, which exploits the generalized lp-norm as the

Over the last decade, the sparse representation model has led to remarkable results in numerous signal and image processing applications. To incorporate the inherent structure of the data and account for the fact that not all support patterns are equally likely, this model was enriched by enforcing various structural sparsity patterns. One plausible such extension of classic sparse coding, instigated

Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular, the probit algorithm, level set and kriging methods. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian

We consider the bilinear inverse problem of recovering two vectors, x and w, in RL from their entrywise product. For the case where the vectors have known signs and belong to known subspaces, we introduce the convex program BranchHull, which is posed in the natural parameter space that does not require an approximate solution or initialization in order to be stated or solved. Under the structural assumptions

Holographic representations of data encode information in packets of equal importance that enable progressive recovery. The quality of recovered data improves as more and more packets become available. This progressive recovery of the information is independent of the order in which packets become available. Such representations are ideally suited for distributed storage and for the transmission of

This paper considers the problem of phase retrieval, where the goal is to recover a signal z∈Cn from the observations yi=|ai⁎z|, i=1,2,⋯,m. While many algorithms have been proposed, the alternating minimization algorithm is still one of the most commonly used and the simplest methods. Existing works have proved that when the observation vectors {ai}i=1m are sampled from a complex normal distribution

This paper discusses isotropic sparse regularization for a random field on the unit sphere S2 in R3, where the field is expanded in terms of a spherical harmonic basis. A key feature is that the norm used in the regularization term, a hybrid of the ℓ1 and ℓ2-norms, is chosen so that the regularization preserves isotropy, in the sense that if the observed random field is strongly isotropic then so too