A Low-Rank Approach to Off-The-Grid Sparse Deconvolution

TL;DR

This paper introduces a scalable low-rank method for off-the-grid sparse deconvolution that efficiently recovers sparse signals from convolutive measurements, outperforming traditional SDP approaches in large-scale settings.

Contribution

It presents a low-rank penalized formulation and a non-convex conditional gradient algorithm that exploits problem structure for efficient large-scale sparse deconvolution.

Findings

Algorithm converges in exactly r steps, where r is the number of Diracs.

Complexity per iteration is O(f_c^d log f_c), scalable for large problems.

Numerical simulations show promising convergence and accuracy.

Abstract

We propose a new solver for the sparse spikes deconvolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite (SDP) relaxations of the total variation (the total mass of the absolute value of the measure) minimization problem. The direct resolution of this SDP is however intractable for large scale settings, since the problem size grows as $f_c^{2d}$ where $f_c$ is the cutoff frequency of the filter and $d$ the ambient dimension. Our first contribution introduces a penalized formulation of this semidefinite lifting, which has low-rank solutions. Our second contribution is a conditional gradient optimization scheme with non-convex updates. This algorithm leverages both the low-rank and the convolutive structure of the problem, resulting in an $O(f_c^d \log f_c)$ complexity per iteration. Numerical simulations are promising and show that the algorithm converges in exactly $r$ steps, $r$ being the number of Diracs composing the solution.