A Compact Formulation for the $\ell_{2,1}$ Mixed-Norm Minimization Problem

TL;DR

This paper introduces a compact reformulation of the $,1$ mixed-norm minimization problem for joint sparse signal recovery, reducing problem size and enabling gridless parameter estimation with theoretical equivalences.

Contribution

It derives a compact reformulation of the $,1$ mixed-norm minimization, linking it to atomic-norm minimization and providing efficient algorithms for various sampling scenarios.

Findings

Significant reduction in problem size for joint sparse recovery.

Establishment of exact equivalence between mixed-norm and atomic-norm minimization.

Development of low-complexity algorithms for irregular sampling cases.

Abstract

Parameter estimation from multiple measurement vectors (MMVs) is a fundamental problem in many signal processing applications, e.g., spectral analysis and direction-of- arrival estimation. Recently, this problem has been address using prior information in form of a jointly sparse signal structure. A prominent approach for exploiting joint sparsity considers mixed-norm minimization in which, however, the problem size grows with the number of measurements and the desired resolution, respectively. In this work we derive an equivalent, compact reformulation of the $\ell_{2,1}$ mixed-norm minimization problem which provides new insights on the relation between different existing approaches for jointly sparse signal reconstruction. The reformulation builds upon a compact parameterization, which models the row-norms of the sparse signal representation as parameters of interest, resulting in a significant reduction of the MMV problem size. Given the sparse vector of row-norms, the jointly sparse signal can be computed from the MMVs in closed form. For the special case of uniform linear sampling, we present an extension of the compact formulation for gridless parameter estimation by means of semidefinite programming. Furthermore, we derive in this case from our compact problem formulation the exact equivalence between the $\ell_{2,1}$ mixed-norm minimization and the atomic-norm minimization. Additionally, for the case of irregular sampling or a large number of samples, we present a low complexity, grid-based implementation based on the coordinate descent method.