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Abstract In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix-vector multiplication, peak finding, and clustering. [ABSTRACT FROM AUTHOR]

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

In this paper we solve an open problem posed in [10] related to the representation of Positive Operator-Valued Measures by means of tight probabilistic frames in R d. Also, we investigate how far is the closest probabilistic tight frame from a given probability measure where the distance used is the quadratic Wasserstein metric W 2 used for optimal transportation problem for measures. In particular, we study the optimization problem I (μ , K) : = inf ν ∈ T (K) ⁡ W 2 2 (μ , ν) , where T (K) is the set of all probabilistic tight frames whose supports are contained in K = R d or K = S d − 1. This problem is solved and its optimum is given when the mean vector of μ is zero. In the other cases, we give concise upper and lower bounds for I (μ , K). [ABSTRACT FROM AUTHOR]

Matrix completion involves recovering a matrix from a subset of its entries by utilizing interdependency between the entries, typically through low rank structure. Despite matrix completion requiring the global solution of a non-convex objective, there are many computationally efficient algorithms which are effective for a broad class of matrices. In this paper, we introduce an alternating steepest descent algorithm (ASD) and a scaled variant, ScaledASD, for the fixed-rank matrix completion problem. Empirical evaluation of ASD and ScaledASD on both image inpainting and random problems show they are competitive with other state-of-the-art matrix completion algorithms in terms of recoverable rank and overall computational time. In particular, their low per iteration computational complexity makes ASD and ScaledASD efficient for large size problems, especially when computing the solutions to moderate accuracy such as in the presence of model misfit, noise, and/or as an initialization strategy for higher order methods. A preliminary convergence analysis is also presented. [ABSTRACT FROM AUTHOR]

In this paper we solve support vector machines in reproducing kernel Banach spaces (RKBSs) instead of the traditional methods in reproducing kernel Hilbert spaces (RKHSs). Using the orthogonality of semi-inner-products of RKBSs, we can obtain the finite-dimensional representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can use Fourier transform techniques to introduce the concept of reproduction in a generalized native space such that it becomes a reproducing kernel Banach space, which can even be embedded into Sobolev spaces. Moreover, its reproducing kernel is associated with a positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly and finite-dimensionally in terms of the positive definite functions, and their finite numbers of suitable parameters can be computed by the fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Matérn functions (Sobolev splines) such that their support vector machine solutions even are computable efficiently. Moreover, each of their reproducing bases includes information from multiple training data points. These kernel-based algorithms give a fresh numerical tool for support vector classifiers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an extension of Pauli's phase retrieval problem. The original problem asks whether a function u is uniquely determined by its modulus | u | and the modulus of its Fourier transform | F u | up to a constant phase factor. Here we extend this problem by considering the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if u and v are such that fractional Fourier transforms of order α have same modulus | F α u | = | F α v | for some set τ of α 's, then v is equal to u up to a constant phase factor. The set τ depends on some extra assumptions either on u or on both u and v . Cases considered here are u , v of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the set τ may even be reduced to a single point ( i.e. one fractional Fourier transform may suffice for uniqueness in the problem). [ABSTRACT FROM AUTHOR]

The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurements. We first investigate the minimal number of measurements for the success of the recovery of sparse signals from the magnitude of samples. We completely settle the minimality question for the real case and give a bound for the complex case. We then study the recovery performance of the ℓ 1 minimization for the sparse phase retrieval problem. In particular, we present the null space property which, to our knowledge, is the first sufficient and necessary condition for the success of ℓ 1 minimization for k -sparse phase retrieval. [ABSTRACT FROM AUTHOR]