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

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

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

In this paper, we introduce the new class of twisted B-splines and study some properties of these B-splines. We also investigate the system of twisted translates and the wavelets corresponding to these twisted B-splines.

We study a geometric property related to spherical hyperplane tessellations in Rd. We first consider a fixed x on the Euclidean sphere and tessellations with M≫d hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of dlog⁡(d)log⁡(M)

We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine

This work provides new results for the analysis of random sequences in terms of ℓp-compressibility. The results characterize the degree in which a random sequence can be approximated by its best k-sparse version under different rates of significant coefficients (compressibility analysis). In particular, the notion of strong ℓp-characterization is introduced to denote a random sequence that has a well-defined

In this paper, we consider a class of structured fractional minimization problems where the numerator part of the objective is the sum of a convex function and a Lipschitz differentiable (possibly) nonconvex function, while the denominator part is a convex function. By exploiting the structure of the problem, we propose a first-order algorithm, namely, a proximal-gradient-subgradient algorithm with

In [3], Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of

In this paper, we study the stability of phase retrieval problems via a family of locally stable phase retrieval frame measurements in Banach spaces, which we call “locally stable and conditionally connected” (LSCC) measurement schemes. For any signal f in the Banach space, we associate it with a weighted graph Gf, defined by the LSCC measurement scheme, and show that the phase retrievability of the

Following recent interest by the community, the scaling of the minimal singular value of a Vandermonde matrix with nodes forming clusters on the length scale of Rayleigh distance on the complex unit circle is studied. Using approximation theoretic properties of exponential sums, we show that the decay is only single exponential in the size of the largest cluster, and the bound holds for arbitrary small

Homomorphic sensing is a recent algebraic-geometric framework that studies the unique recovery of points in a linear subspace from their images under a given collection of linear maps. It has been successful in interpreting such a recovery in the case of permutations composed by coordinate projections, an important instance in applications known as unlabeled sensing, which models data that are out

Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent articles, including pseudodifferential, Fourier integral and metaplectic operators, with applications to the analysis of Schrödinger-type evolution equations. It has been proved that such operators are approximately diagonalized by Gabor wave packets. While the latter are expected

Let G⊂L2(R) be the subspace spanned by a Gabor Riesz sequence (g,Λ) with g∈L2(R) and a lattice Λ⊂R2 of rational density. It was shown recently that if g is well-localized both in time and frequency, then G cannot contain any time-frequency shift π(z)g of g with z∈R2∖Λ. In this paper, we improve the result to the quantitative statement that the L2-distance of π(z)g to the space G is equivalent to the

We study the ratio of ℓ1 and ℓ2 norms (ℓ1/ℓ2) as a sparsity-promoting objective in compressed sensing. We first propose a novel criterion that guarantees that an s-sparse signal is the local minimizer of the ℓ1/ℓ2 objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement

This paper investigates the nonparametric regression problem using SVMs with anisotropic Gaussian RBF kernels. Under the assumption that the target functions are resided in certain anisotropic Besov spaces, we establish the almost optimal learning rates, more precisely, optimal up to some logarithmic factor, presented by the effective smoothness. By taking the effective smoothness into consideration

In the manifold setting, we provide a series of spectral convergence results quantifying how the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator in the L∞ sense. Based on these results, convergence of the proposed heat kernel approximation algorithm, as well as the convergence rate, to the exact heat kernel is guaranteed

The attenuated X-ray transform arises from the image reconstruction in single photon emission computed tomography. The theory of attenuated X-ray transforms is so far incomplete and many questions remain open. In this paper, we are pursuing the inversion of the attenuated X-ray transforms with nonnegative varying attenuation functions μ. By the symmetric attenuated X-ray transform Aμ on plane and the

We show that orthonormality of a discrete Gabor bases on Cn hinges heavily on the following pattern of its support set Γ⊂Zn×Zn: (i) Γ is itself a subgroup of order n, or (ii) Γ is the quotient of such a subgroup, i.e., there exists an order n subgroup H◁Zn×Zn such that Γ takes precisely one element from each coset of H (i.e., Zn×Zn=H×Γ). If n is a prime number, then Γ satisfying (i) automatically implies

We propose and study kernel conjugate gradient methods (KCGM) with random projections for least-squares regression over a separable Hilbert space. Considering two types of random projections generated by randomized sketches and Nyström subsampling, we prove optimal statistical results with respect to variants of norms for the algorithms under a suitable stopping rule. Particularly, our results show

The objective of this paper is to introduce an innovative approach for the recovery of non-stationary signal components with possibly crossover instantaneous frequency (IF) curves from a multi-component blind-source signal. The main idea is to incorporate a chirp rate parameter with the time-scale continuous wavelet-like transformation, by considering the quadratic phase representation of the signal

While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in time-frequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in CN by their FROG measurements. After establishing the ambiguity

In this paper we first prove the existence of the n-best approximation in terms of the parameterized Szegö kernels in the stochastic complex Hardy space of the unit disc. It is a generalization to random signals of the corresponding result for the Hardy space of the disc, and has applications in signal analysis. The result may be generalized to the randomized polydisc case with potential applications

Matrix completion is about recovering a matrix from its partial revealed entries, and it can often be achieved by exploiting the inherent simplicity or low dimensional structure of the target matrix. For instance, a typical notion of matrix simplicity is low rank. In this paper we study matrix completion based on another low dimensional structure, namely the low rank Hankel structure in the Fourier

In this work, we study the spectral property of generalized Sierpinski-type self-affine measures μM,D on R2 generated by an expanding integer matrix M∈M2(Z) with det⁡(M)∈3Z and a non-collinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t} with α1β2−α2β1∈3Z. We give the sufficient and necessary conditions for μM,D to be a spectral measure, i.e., there exists a countable subset Λ⊂R2 such that E(Λ)={e2πi〈λ

The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in ℓ2(Z) which are strictly bandlimited to a frequency band [−W,W] and maximally concentrated in a time interval {0,…,N−1}. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in CN whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency

We introduce almost diagonal matrices in the setting of (anisotropic) discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation spaces, and it is shown that the class of almost diagonal matrices is closed under matrix multiplication. We then connect the results to the continuous setting and show that the “change of frame” matrix for a pair of time-frequency frames, with suitable

This paper is devoted to the Lp(R) theory of the fractional Fourier transform (FRFT) for 1≤p<2. In view of the special structure of the FRFT, we study FRFT properties of L1 functions, via the introduction of a suitable chirp operator. However, in the L1(R) setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion

Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modeled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable

The present work investigates the segmentation of textures by formulating it as a strongly convex optimization problem, aiming to favor piecewise constancy of fractal features (local variance and local regularity) widely used to model real-world textures in numerous applications very different in nature. Two objective functions combining these two features are compared, referred to as joint and coupled

In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain Ω of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in Ω. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel

We present a construction of C1 piecewise quadratic hierarchical bases of Lagrange type on arbitrary polygonal domains Ω⊂R2. Properly normalized, these bases are Riesz bases for Sobolev spaces Hs(Ω), with s∈(1,52). The method is applicable to arbitrary initial triangulations of polygonal domains, and does not require a checkerboard quadrangulation needed for earlier C1 cubic hierarchical Lagrange bases

In this article, the authors establish the wavelet characterization of Besov and Triebel–Lizorkin spaces on a given space (X,d,μ) of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel–Lizorkin sequence spaces on X, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal

This paper proposes a new methodology for solving the well-known rank aggregation problem from pairwise comparisons using low-rank matrix completion. Partial and noisy data of pairwise comparisons is first transformed into a matrix form. We then use tools from matrix completion, which has served as a major component in the low-rank based completion solution for the Netflix challenge, to construct the

We propose a distributed learning algorithm for least squares regression in reproducing kernel Hilbert spaces (RKHSs) generated by flexible Gaussian kernels, based on a divide-and-conquer strategy. Our study demonstrates that Gaussian kernels with flexible variances greatly improve the learning performance of distributed algorithms generated by a fixed Gaussian. Under some mild conditions, we establish

In this paper, we extend the diffusion maps algorithm on a family of heat kernels that are either local (having exponential decay) or nonlocal (having polynomial decay), arising in various applications. For example, these kernels have been used as a regularizer in various supervised learning tasks for denoising images. Importantly, these heat kernels give rise to operators that include (but are not

Concentration of frequency and time (ConceFT) is a generalized multitaper algorithm introduced to analyze complicated non-stationary time series. To avoid the randomness in the original ConceFT algorithm, we apply the novel complex spherical design technique to standardize ConceFT, which we coin CQU-ConceFT. The proposed CQU-ConceFT is applied to visualize the spindle structure in the electroencephalogram

Variational regularization models are one of the popular and efficient approaches for image restoration. The regularization functional in the model carries prior knowledge about the image to be restored. The prior knowledge, in particular for natural images, are the first-order (i.e. variance in luminance) and second-order (i.e. contrast and texture) information. In this paper, we propose a model for

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). The operator Wτ is skew-adjoint, and thus can be represented by a

We consider the problem of classification of points sampled from an unknown probability measure on a Euclidean space. We study the question of querying the class label at a very small number of judiciously chosen points so as to be able to attach the appropriate class label to every point in the set. Our approach is to consider the unknown probability measure as a convex combination of the conditional

In this paper, we theoretically investigate the low-rank matrix recovery problem in the context of the unconstrained regularized nuclear norm minimization (RNNM) framework. Our theoretical findings show that, the RNNM method is able to provide a robust recovery of any matrix X (not necessary to be exactly low-rank) from its few noisy measurements b=A(X)+n with a bounded constraint ‖n‖2≤ϵ, provided

Dimensionality reduction methods are designed to overcome the ‘curse of dimensionality’ phenomenon that makes the analysis of high dimensional big data difficult. Many of these methods are based on principal component analysis which is statistically driven and do not directly address the geometry of the data. Thus, machine learning tasks, such as classification and anomaly detection, may not benefit

Orthogonal and biorthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. On the other hand, a lot of problems in applications such as images and solutions of differential equations are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted

In nature and the technology world, acquired signals and time series are usually affected by multiple complicated factors and appear as multi-component non-stationary modes. In many situations it is necessary to separate these signals or time series to a finite number of mono-components to represent the intrinsic modes and underlying dynamics implicated in the source signals. Recently the synchrosqueezed

Rician noise is a common noise that naturally appears in Magnetic Resonance Imaging (MRI) images. Low rank matrix approximation approaches have been widely used in image processing, which takes advantage of the non-local self-similarity between patches in a natural image. The weighted nuclear norm minimization method as a low rank matrix approximation approach has shown to be an effective approach

The covariance of a stationary process X is diagonalized by a Fourier transform. It does not take into account the complex Fourier phase and defines Gaussian maximum entropy models. We introduce a general family of phase harmonic covariance moments, which rely on complex phases to capture non-Gaussian properties. They are defined as the covariance of Hˆ(LX), where L is a complex linear operator and

In this paper, we propose a unified theoretical and practical spherical approximation framework for functional inverse problems on the hypersphere Sd−1. More specifically, we consider recovering spherical fields directly in the continuous domain using functional penalised basis pursuit problems with generalised total variation (gTV) regularisation terms. Our framework is compatible with various measurement

Modulation spaces Mp,qs were introduced by Feichtinger [11] in 1983. Bényi and Oh [2] defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possible applications in PDE. By carefully investigating the scaling properties of modulation spaces and their connections with Bényi and Oh's modulation spaces, we introduce the scaling limit versions

Generally, the phase retrieval problem can be viewed as the reconstruction of a function/signal from only the magnitude of linear measurements. These measurements can be, for example, the Fourier transform of the density function. Computationally the phase retrieval problem is very challenging. Many algorithms for phase retrieval are based on i.i.d. Gaussian random measurements. However, Gaussian random

The paper studies predicting problems for continuous time signals with the Fourier transforms vanishing with a certain rate at a single point. It shows that, with some linear causal predictors, anticausal convolution integrals over future times for these processes can be approximated by causal convolutions over past times. The corresponding predicting kernels are time invariant, and they are presented

In this paper we study sharp constants in inequalities of the following form‖x(k)(⋅)‖Lq(R)≤K‖Fx(⋅)‖Lp(R)α‖x(n)(⋅)‖Lr(R)β, where Fx(⋅) is the Fourier transform of x(⋅). The sharp value of K in the general case (that is, for all n∈N and 0≤k

The Haar function is extended to a family of minimum-supported cardinal spline-wavelets ψm,n, with any desired polynomial order m and arbitrarily high order n of vanishing moments, for the purpose of carrying out our strategy of continuous wavelet transform (CWT) thresholding to recover all “active” sub-signals, along with their instantaneous frequencies (IFs), from a blind-source composite signal