Projected gradient descent for non-convex sparse spike estimation

We propose a new algorithm for sparse spike estimation from Fourier measurements. Based on theoretical results on non-convex optimization techniques for off-the-grid sparse spike estimation, we present a projected gradient descent algorithm coupled with a spectral initialization procedure. Our algorithm permits to estimate the positions of large numbers of Diracs in 2d from random Fourier measurements. We present, along with the algorithm, theoretical qualitative insights explaining the success of our algorithm. This opens a new direction for practical off-the-grid spike estimation with theoretical guarantees in imaging applications

Iterative Discretization of Optimization Problems Related to Superresolution

International audienceWe study an iterative discretization algorithm for solving optimization problems regularized by the total variation norm over the space M(Ω) of Radon measures on a bounded subset Ω of R d. Our main motivation to study this problem is the recovery of sparse atomic measures from linear measurements. Under reasonable regularity conditions, we arrive at a linear convergence rate guarantee

PROJECTED GRADIENT DESCENT FOR NON-CONVEX SPARSE SPIKE ESTIMATION

We propose an algorithm to perform sparse spike estimation from Fourier measurements. Based on theoretical results on non-convex optimization techniques for off-the-grid sparse spike estimation, we present a simple projected descent algorithm coupled with an initialization procedure. Our algorithm permits to estimate the positions of large numbers of Diracs in 2d from random Fourier measurements. This opens the way for practical estimation of such signals for imaging applications as the algorithm scales well with respect to the dimensions of the problem. We present, along with the algorithm, theoretical qualitative insights explaining the success of our algorithm

Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric $K$-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known {\em a priori} to lie in certain frequency bands. Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin

Déconvolution parcimonieuse sans grille: une méthode de faible rang

International audienceOn s'intéresse à la résolution numérique du problème de déconvolution sans grille pour des mesures de Radon discrètes. Une approche courante consiste à introduire des relaxations semidéfinies positives (SDP) du problème variationnel associé, qui correspond ici à un problème de minimisation de variation totale. Cependant, pour des signaux de dimension supérieure à 1, les méthodes usuelles de points intérieurs sont peu efficaces pour résoudre le programme SDP correspondant, la taille de celui-ci étant de l'ordre de fc^{2d} où fc désigne la fréquence de coupure du filtre et d la dimension du signal. Nous introduisons en premier lieu une version pénalisée de la formulation SDP, dont les solutions sont de faible rang. Nous proposons ensuite un schéma numérique basé sur l'algorithme de Frank-Wolfe, capable d'exploiter efficacement d'une part cette propriété de faible rang, d'autre part l'aspect convolutif du problème; notre méthode atteint ainsi un coût de l'ordre de O(fc^d log fc) par itération. Nos simulations sont prometteuses, et montrent que l'algorithme converge en k étapes, k étant le nombre de Diracs dans la solution