We present a fast method for evaluating expressions of the formuj=∑i=1,i≠jnαixi−xj,forj=1,…,n, where αi are real numbers, and xi are points in a compact interval of R. This expression can be viewed as representing the electrostatic potential generated by charges on a line in R3. While fast algorithms for computing the electrostatic potential of general distributions of charges in R3 exist, in a number

We introduce two shearlet-based Ginzburg–Landau energies, based on the continuous and the discrete shearlet transform. The energies result from replacing the elastic energy term of a classical Ginzburg–Landau energy by the weighted L2-norm of a shearlet transform. The asymptotic behaviour of sequences of these energies is analysed within the framework of Γ-convergence and the limit energy is identified

We propose a new solution to the blind source separation problem that factors mixed time-series signals into a sum of spatiotemporal modes, with the constraint that the temporal components are intrinsic mode functions (IMF's). The key motivation is that IMF's allow the computation of meaningful Hilbert transforms of non-stationary data, from which instantaneous time-frequency representations may be

We present a new infinite family of full spark equal norm tight frames in finite dimensions arising from a unitary group representation, where the underlying group is the semi-direct product of a cyclic group by a group of automorphisms. The only previously known algebraically constructed infinite families were the harmonic, Gabor and Dihedral frames. Our construction hinges on a theorem that requires

We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks

Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the L2 space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series

We study the theoretical underpinning of a robust empirical risk minimization (RERM) scheme which has been finding numerous successful applications across various data science fields owing to its robustness to outliers and heavy-tailed noises. The specialties of RERM lie in its nonconvexity and that it is induced by a loss function with an integrated scale parameter trading off the robustness and the

In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in Cn are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep

Multiresolution analysis (MRA) on a compact abelian group G has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on Lp(G), 1≤p<∞. With the help of the scaling sequence we construct an orthonormal wavelet basis of L2(G).

In this paper we continue our work in [13] where the minimization problem I(μ):=infν∈T⁡W22(μ,ν),T being the set of probabilistic tight frames in Rd and W2 the quadratic Wasserstein metric for measures, was solved in the particular case when the mean vector of the probabilistic frame μ is zero. The present work solves this problem in the general case, and in addition, establishes the uniqueness of the

The field of compressed sensing has become a major tool in high-dimensional analysis, with the realization that vectors can be recovered from relatively very few linear measurements as long as the vectors lie in a low-dimensional structure, typically the vectors that are zero in most coordinates with respect to a basis. However, there are many applications where we instead want to recover vectors that

We study the upper and the lower densities of complete and minimal Gabor Gaussians systems. In contrast to the classical lattice case when they are both equal to 1, we prove that the lower density may reach 0 while the upper density may vary at least from 1π to e. In the case when the upper density exceeds 1, we establish a sharp inequality relating the upper and the lower densities.

Finding a good regularization parameter for Tikhonov regularization problems is a though yet often asked question. One approach is to use leave-one-out cross-validation scores to indicate the goodness of fit. This utilizes only the noisy function values but, on the downside, comes with a high computational cost. In this paper we present a general approach to shift the main computations from the function

Several strategies have been applied for the recovery of the missing parts in an image, with recovery performance depending significantly on the image type and the geometry of missing data. To provide a deeper understanding of such image restoration problem, King et al. recently introduced a rigorous multiscale analysis framework and proved that a shearlet based inpainting approach outperforms methods

In this paper, we consider an unconstrained ℓ2,q minimization for group sparse signal recovery. For this nonconvex and non-Lipschitz problem, we mainly focus on its local minimizers. Firstly, a uniform lower bound for nonzero groups of the local minimizers is presented. Secondly, under group restricted isometry property (GRIP) assumption, we provide a global recovery bound for points in a sublevel

For any positive real number p, the p-frame potential of N unit vectors X:={x1,…,xN}⊂Rd is defined as FPp,N,d(X)=∑i≠j|〈xi,xj〉|p. In this paper, we focus on this quantity for N=d+1 points and establish uniqueness of minimizers of FPp,d+1,d for all p∈(0,2). Our results completely solve the minimization problem of the p-frame potential when N=d+1, confirming a conjecture posed by Chen, Gonzales, Goodman

The definition of graph Fourier transform is a fundamental issue in graph signal processing. Conventional graph Fourier transform is defined through the eigenvectors of the graph Laplacian matrix, which minimize the ℓ2 norm signal variation. In this paper, we propose a generalized definition of graph Fourier transform based on the ℓ1 norm variation minimization. We obtain a necessary condition satisfied

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First

In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for

This work investigates the parameter estimation performance of super-resolution line spectral estimation using atomic norm minimization. The focus is on analyzing the algorithm's accuracy of inferring the frequencies and complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is reasonably high and the true frequencies are separated by O(1n), the atomic norm estimator is shown to

We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves

Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image

Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson

Sampling a signal below the Shannon–Nyquist rate causes aliasing, meaning different frequencies to become indistinguishable. It is also well-known that recovering spectral information from a signal using a parametric method can be ill-posed or ill-conditioned and therefore should be done with caution. We present an exponential analysis method to retrieve high-resolution information from coarse-scale

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.

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

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. 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 stable signal estimation from undersampled, noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, a novel recovery guarantee for the ℓ1-analysis basis pursuit is derived, enabling accurate predictions of its sample complexity. The bounds on the number of required measurements explicitly depend on the

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)|

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

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

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

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

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

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