Necessary and sufficient conditions are presented for several families of planar curves to form a set of stable sampling for the Bernstein space BΩ over a convex set Ω⊂R2. These conditions ‘essentially’ describe the mobile sampling property of these families for the Paley-Wiener spaces PWΩp,1≤p<∞.

In many applications, it is necessary to retrieve the sub-signal building blocks of a multi-component signal, which is usually non-stationary in real-world and real-life applications. Empirical mode decomposition (EMD), synchrosqueezing transform (SST), signal separation operation (SSO), and iterative filtering decomposition (IFD) have been proposed and developed for this purpose. However, these computational

Deep neural networks (DNNs) are quantized for efficient inference on resource-constrained platforms. However, training deep learning models with low-precision weights and activations involves a demanding optimization task, which calls for minimizing a stage-wise loss function subject to a discrete set-constraint. While numerous training methods have been proposed, existing studies for full quantization

Wavelet quantile normalization (WQN) is a nonparametric algorithm designed to remove transient artifacts from single-channel EEG in real-time. EEG monitoring machines suspend their output when artifacts in the signal are detected. Removing unpredictable EEG artifacts can improve the continuity of monitoring. We analyse here the WQN algorithm which consists in transporting wavelet coefficient distributions

Recently, the Randomized Kaczmarz algorithm (RK) draws much attention because of its low computational complexity and less requirement on computer memory. Many existing results on analysis focus on the behavior of RK in Euclidean spaces, and typically derive exponential converge rates with the base tending to one, as the condition number of the system increases. The dependence on the condition number

We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex ℓ1 penalty and many of its non-convex surrogates that have been well studied in the literature. We

The collection of d×N complex matrices with prescribed column norms and singular values forms an algebraic variety, which we refer to as a frame space. Elements of frame spaces—i.e., frames—are used to give robust signal representations, so that geometrical properties of frame spaces are of interest to the signal processing community. This paper is concerned with the question: what is the probability

Three families of super-resolution (SR) wavelets Ψv,ng(x), Ψu,n,ms(x) and Ψw,n,mds(x), to be called Gaussian SR (GSR), spline SR (SSR) and dual-spline SR (DSSR) wavelets, respectively, are introduced in this paper for resolving the super-resolution problem of recovering any point-mass h(y)=∑ℓ=1Lcℓδ(y−σℓ), with |σℓ−σk|≥η for ℓ≠k, σℓ≠0, and |cℓ|>η⁎ for all ℓ,k=1,…,L, where η>0 and η⁎>0 are allowed to

We give a full description of complete interpolating sequences for the shift-invariant space generated by the Gaussian. As a consequence, we rederive the known density conditions for sampling and interpolation.

We study the spectral convergence of graph Laplacians to the Laplace-Beltrami operator when the kernelized graph affinity matrix is constructed from N random samples on a d-dimensional manifold in an ambient Euclidean space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove eigen-convergence with rates as

Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon. Theoretically, graphons are limit objects of converging sequences of graphs. Our work extends previous research proposing a common scheme for signal analysis of

Hard-thresholding-based algorithms have seen various advantages for sparse optimization in controlling the sparsity and allowing for fast computation. Recent research shows that when techniques of the Newton-type methods are integrated, their numerical performance can be improved surprisingly. This paper develops a gradient projection Newton pursuit algorithm that mainly adopts the hard-thresholding

We provide a sufficient condition for sets of mobile sampling in terms of the surface density of the set.

Convex ℓ1 regularization using an infinite dictionary of neurons has been suggested for constructing neural networks with desired approximation guarantees, but can be affected by an arbitrary amount of over-parametrization. This can lead to a loss of sparsity and result in networks with too many active neurons for the given data, in particular if the number of data samples is large. As a remedy, in

It is known that, in general, an AP-frame is an L2(R)-frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system {g(t−k)eil(t−k),l∈L=ω0Z,k∈K=t0Z} to be an L2(R)-Frame in terms of Gaussian stationary random processes. In addition, if X=(X(t))t∈R is a wide sense stationary random process, we study density conditions for

The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces

For any dilation matrix with integral entries A∈Rd×d, d≥1, we construct two families of Parseval wavelet frames in L2(Rd). Both families have compact support and any desired number of vanishing moments. The first family has |det⁡A| generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension

Divergence-free interpolation has been extensively studied and widely used in approximating vector-valued functions that are divergence-free. However, so far the literature contains no treatment of divergence-free quasi-interpolation. The aims of this paper are two-fold: to construct an analytically divergence-free quasi-interpolation scheme and to derive its simultaneous approximation orders to both

In this paper, we propose a model for parallel magnetic resonance imaging (pMRI) reconstruction, regularized by a carefully designed tight framelet system, that can lead to reconstructed images with much less artifacts in comparison to those from existing models. Our model is motivated from the observations that each receiver coil in a pMRI system is more sensitive to the specific object nearest to

The framework of this article is to introduce a new efficient Blind Source Separation (BSS) method that handles mixtures of noise-contaminated independent / dependent sources. In order to achieve that, one can minimize a criterion that fuses a separating part, based on Kullback–Leibler divergence to set apart the observed mixtures of either dependent or independent sources, with a regularization part

Hirschman and Widder introduced a class of Pólya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a Pólya frequency function only if the polynomial is a homothety

For the interpolation of graph signals with generalized shifts of a graph basis function (GBF), we introduce the concept of positive definite functions on graphs. This concept merges kernel-based interpolation with spectral theory on graphs and can be regarded as a graph analog of radial basis function interpolation in Euclidean spaces or spherical basis functions. We provide several descriptions of

The objective of this study is to present a shape-preserving non-linear subdivision scheme generalizing the exponential B-spline of degree 3, which is a piecewise exponential polynomial with the same support as the cubic B-spline. The subdivision of the exponential B-spline has a crucial limitation in that it can reproduce at most two exponential polynomials, yielding the approximation order two. Also

The efficiency of modern computer graphics allows us to explore collections of space curves simultaneously with “drag-to-rotate” interfaces. This inspires us to replace “scatterplots of points” with “scatterplots of curves” to simultaneously visualize relationships across an entire dataset. Since spaces of curves are infinite dimensional, scatterplots of curves avoid the “lossy” nature of scatterplots

Group-sparsity is a common low-complexity signal model with widespread application across various domains of science and engineering. The recovery of such signal ensembles from compressive measurements has been extensively studied in the literature under the assumption that measurement operators are modeled as densely populated random matrices. In this paper, we turn our attention to an acquisition

We study zeroth-order optimization for convex functions where we further assume that function evaluations are unavailable. Instead, one only has access to a comparison oracle, which given two points x and y returns a single bit of information indicating which point has larger function value, f(x) or f(y). By treating the gradient as an unknown signal to be recovered, we show how one can use tools from

We consider the problem of characterizing the ‘duality gap’ between sparse synthesis- and cosparse analysis-driven signal models through the lens of spectral graph theory, in an effort to comprehend their precise equivalencies and discrepancies. By detecting and exploiting the inherent connectivity structure, and hence, distinct set of properties, of rank-deficient graph difference matrices such as

Nonlinear wavelet-based estimators for spectral densities of non-Gaussian linear processes are considered. The convergence rates of mean integrated squared error (MISE) for those estimators over a large range of Besov function classes are derived, and it is shown that those rates are identical to minimax lower bounds in standard nonparametric regression model within a logarithmic term. Thus, those

In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of

The reason we wrote this note is twofold. First, in contrast to the (now) classical rectangular lattices αZ × βZ, not much is known about irregular ones Λ×M . The recent breakthrough related to semiregular lattices of the form Λ×βZ has been achieved in [1], where the authors considered the Gabor frames, generated by Gaussian totally positive functions of finite type. We also refer the reader to [1]

Group synchronization asks to recover group elements from their pairwise measurements. It has found numerous applications across various scientific disciplines. In this work, we focus on orthogonal and permutation group synchronization which are widely used in computer vision such as object matching and structure from motion. Among many available approaches, the spectral methods have enjoyed great

We describe a method for the numerical evaluation of the angular prolate spheroidal wave functions of the first kind of order zero. It is based on the observation that underlies the WKB method, namely that many second order differential equations admit solutions whose logarithms can be represented much more efficiently than the solutions themselves. However, rather than exploiting this fact to construct

In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the A-Wigner distribution defined by WA(f)=μ(A)(f⊗f¯), where A is a 4d×4d symplectic matrix and μ(A) is an associate metaplectic operator. Basic examples are given by the so-called τ-Wigner distributions. Such representations provide a new characterization

This paper addresses theory and applications of ℓp-based Laplacian regularization in semi-supervised learning. The graph p-Laplacian for p>2 has been proposed recently as a replacement for the standard (p=2) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate. In the first part of the paper we prove new discrete to continuum convergence

We prove explicit lower bounds for the smallest singular value and upper bounds for the condition number of rectangular, multivariate Vandermonde matrices with scattered nodes on the complex unit circle. Analogously to the Shannon-Nyquist criterion, the nodes are assumed to be separated by a constant divided by the used polynomial degree. If this constant grows linearly with the spatial dimension,

The problem of recovering a signal x∈Rn from a set of magnitude measurements yi=|〈ai,x〉|,i=1,…,m is referred as phase retrieval, which has many applications in fields of physical sciences and engineering. In this paper we show that the smoothed amplitude flow based model for phase retrieval has benign geometric structure under the optimal sampling complexity. In particular, we show that when the measurements

The activation function deployed in a deep neural network has great influence on the performance of the network at initialisation, which in turn has implications for training. In this paper we study how to avoid two problems at initialisation identified in prior works: rapid convergence of pairwise input correlations, and vanishing and exploding gradients. We prove that both these problems can be avoided

We present a deep-learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging. The algorithm is based on a recently developed digital wavefront set extractor as well as the well-known microlocal canonical relation for the Radon transform. We use the wavefront set information about x-ray data to improve the reconstruction by requiring

We consider the variational problem of cross-entropy loss with n feature vectors on a unit hypersphere in Rd. We prove that when d≥n−1, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that, as n→∞ with fixed d, the minimizing points will distribute uniformly on the hypersphere and show a connection with the frame potential

Compressed sensing studies linear recovery problems under structure assumptions. We introduce a new class of measurement operators, coined hierarchical measurement operators, and prove results guaranteeing the efficient, stable and robust recovery of hierarchically structured signals from such measurements. We derive bounds on their hierarchical restricted isometry properties based on the restricted

We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the smoothness in the spectral Barron space of the underlying function f to be approximated. We show that as the smoothness index s of f increases, shallow neural networks with

We analyze the use of the so-called general regularization scheme in the scenario of unsupervised domain adaptation under the covariate shift assumption. Learning algorithms arising from the above scheme are generalizations of importance weighted regularized least squares method, which up to now is among the most used approaches in the covariate shift setting. We explore a link between the considered

One of the most influential results in neural network theory is the universal approximation theorem [1], [2], [3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems—including

In this paper, we show that the commonly used frame soft shrinkage operator, that maps a given vector x∈RN onto the vector T†SγTx, is already a proximity operator, which can therefore be directly used in corresponding splitting algorithms. In our setting, the frame transform matrix T∈RL×N with L≥N has full rank N, T† denotes the Moore-Penrose inverse of T, and Sγ is the usual soft shrinkage operator

Consider the classical supervised learning problem: we are given data (yi,xi), i≤n, with yi a response and xi∈X a covariates vector, and try to learn a model fˆ:X→R to predict future responses. Random feature methods map the covariates vector xi to a point ϕ(xi) in a higher dimensional space RN, via a random featurization map ϕ. We study the use of random feature methods in conjunction with ridge regression

In many applications, we are given access to noisy modulo samples of a smooth function with the goal being to robustly unwrap the samples, i.e., to estimate the original samples of the function. In a recent work, Cucuringu and Tyagi [11] proposed denoising the modulo samples by first representing them on the unit complex circle and then solving a smoothness regularized least squares problem – the smoothness

The distributions of the k-th largest level at the soft edge scaling limit of Gaussian ensembles are some of the most important distributions in random matrix theory, and their numerical evaluation is a subject of great practical importance. One numerical method for evaluating the distributions uses the fact that they can be represented as Fredholm determinants involving the so-called Airy integral

Clustering algorithms partition a dataset into groups of similar points. The clustering problem is very general, and different partitions of the same dataset could be considered correct and useful. To fully understand such data, it must be considered at a variety of scales, ranging from coarse to fine. We introduce the Multiscale Environment for Learning by Diffusion (MELD) data model, which is a family

We study spanning properties of a family of functions translated along simple model sets. We characterize tight frame and dual frame generators for such irregular translates and we apply the results to Gabor systems. We use the connection between model sets and almost periodic functions and rely strongly on a Poisson summations formula for model sets to introduce the so-called bracket product, which

We propose a new point of view in the study of Fourier analysis on graphs, taking advantage of localization in the Fourier domain. For a signal f on vertices of a weighted graph G with Laplacian matrix L, standard Fourier analysis of f relies on the study of functions g(L)f for some filters g on IL, the smallest interval containing the Laplacian spectrum sp(L)⊂IL. We show that for carefully chosen

Given an image generated by the convolution of point sources with a band-limited function, the deconvolution problem involves reconstructing the source number, positions, and amplitudes. This problem is related to many important applications in imaging and signal processing. It is well known that it is impossible to resolve the sources when they are sufficiently close in practice. Rayleigh investigated

In this paper, we consider the sparse phase retrieval problem, recovering an s-sparse signal x♮∈Rn from m phaseless samples yi=|〈x♮,ai〉| for i=1,…,m. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an efficient second-order algorithm for sparse phase retrieval

The q-th order spectrum is a polynomial of degree q in the entries of a signal x∈CN, which is invariant under circular shifts of the signal. For q≥3, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum (q=3) and the trispectrum (q=4), play a prominent role in various statistical signal

One of key problems in signal reconstruction process with the use of frames is to find a dual frame. Typically, a canonical dual frame is used. However, there are many applications where this choice appears to be unfortunate. Due to that fact, it is necessary to develop a tool, which helps to find a suitable dual frame. In this paper we give a method to find every dual frames. The proposed method is

In this paper, we focus on ℓ1−ℓ2 minimization model, i.e., investigating the nonconvex model:min⁡‖x‖1−‖x‖2s.t.Ax=y and provide a null space property of the measurement matrix A such that a vector x can be recovered from Ax via ℓ1−ℓ2 minimization. The ℓ1−ℓ2 minimization model was first proposed by E.Esser, et al (2013) [8]. As a nonconvex model, it is well known that global minimizer and local minimizer

In this paper, we are concerned with differentially private stochastic gradient descent (SGD) algorithms in the setting of stochastic convex optimization (SCO). Most of the existing work requires the loss to be Lipschitz continuous and strongly smooth, and the model parameter to be uniformly bounded. However, these assumptions are restrictive as many popular losses violate these conditions including

For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problemsupa2,…,aN⁡(infz∈D⁡{Re(F(z)):Im(F(z))=0}). We show that the solution is −14sec2⁡πN+2, and the extremal polynomial1UN′(cos⁡πN+2)∑j=1NUN−j+1′(cos⁡πN+2)Uj−1(cos⁡πN+2)zj is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore