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

Multidimensional Scaling (MDS) is a classical technique for embedding data in low dimensions, still in widespread use today. In this paper we study MDS in a modern setting - specifically, high dimensions and ambient measurement noise. We show that as the ambient noise level increases, MDS suffers a sharp breakdown that depends on the data dimension and noise level, and derive an explicit formula for

This paper presents new theoretical results on sparse recovery guarantees for a greedy algorithm, Orthogonal Matching Pursuit (OMP), in the context of continuous parametric dictionaries. Here, the continuous setting means that the dictionary is made up of an infinite uncountable number of atoms. In this work, we rely on the Hilbert structure of the observation space to express our recovery results

We find a new and simple inversion formula for the 2D Radon transform (RT) with a straight use of the shearlet system and of well-known properties of the RT. Since the continuum theory of shearlets has a natural translation to the discrete theory, we also obtain a computable algorithm that recovers a digital image from noisy samples of the 2D Radon transform which also preserves edges. A very well-known

Framelets derived from refinable (vector) functions via the popular oblique extension principle (OEP) are of interest in both theory and applications. Though OEP can increase vanishing moments of framelet generators to improve sparsity, it has a serious shortcoming for scalar framelets: the associated discrete framelet transform is often not compact and deconvolution is unavoidable. On the other hand

In this note, we solve the dynamical sampling problem for a class of shift-preserving operators L:V→V acting on a finitely generated shift-invariant space V. We find conditions on L and a finite set of functions of V so that the iterations of the operator L on the functions produce a frame generator set of V. This means that the integer translations of the generators form a frame of V.

The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications. In order to deal with high-dimensional problems, efficient sparse fast Fourier transform algorithms are essential tools. The second and third authors have recently proposed a dimension-incremental approach, which only scales almost linear in the number of required sampling values and almost

Bispectral analysis is an effective signal processing tool for investigating interactions between oscillations, and has been adapted to the continuous wavelet transform for time-evolving analysis of open systems. However, one unaddressed question for the wavelet bispectrum is quantification of the bispectral content of an area of scale-scale space. Without this, interpretation of wavelet bispectrum

The first part of the paper proves the conjectures on inequalities in the Schatten p-quasi-norm of matrices. The second part of the paper uses the inequalities for proving a sufficient condition when the Schatten p-quasi-norm minimization can be used for low rank matrix recovery. More precisely, when the restricted isometry constant δ2s<1, there exists a real number p0<1 such that any solution of the

It is known [6], [14], [19] that if Ω⊂Rd belongs to a class of k-tiling domains when translated by a lattice Λ, there exists a Riesz basis of exponentials for L2(Ω) constructed using k translates of the dual lattice Λ⁎. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in [11] to this

We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given M+1 of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance Δ between the two closest point sources is less than 1/M. We introduce a clumps model where the point sources are closely spaced within

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel ϕ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new “exponential of semicircle” kernel ϕ(z)=eβ1−z2

This paper explores essence of the fractional (0

We introduce Ψec(Rn), a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential r-form wavelets with flexible directional localization that provide tight frames for the spaces Ωr(Rn) of forms in R2 and R3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the k-dimensional integral

We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function

The aim of generalized phase retrieval is to recover x∈Fd from the quadratic measurements x⁎A1x,…,x⁎ANx, where Aj∈Hd(F) and F=R or C. In this paper, we study the matrix set A=(Aj)j=1N which has the almost everywhere phase retrieval property. For the case F=R, we show that N≥d+1 generic matrices with prescribed ranks have almost everywhere phase retrieval property. We also extend this result to the

We study the stable recovery of complex k-sparse signals from as few phaseless measurements as possible. The main result is to show that one can employ ℓ1 minimization to stably recover complex k-sparse signals from m≥O(klog⁡(n/k)) complex Gaussian random quadratic measurements with high probability. To do that, we establish that Gaussian random measurements satisfy the restricted isometry property

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 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 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 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 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