Sparse Signal Estimation by Maximally Sparse Convex Optimization

TL;DR

This paper introduces a novel convex optimization method using maximally sparse non-convex penalties, enabling stronger sparsity in signal estimation without the complexities of non-convex optimization.

Contribution

It proposes a new class of penalty functions constrained to keep the overall cost function convex, improving sparsity beyond traditional L1 methods.

Findings

IMSC yields significantly sparser solutions than L1 minimization.

Optimal parameters for penalties are obtained via semidefinite programming.

The approach maintains convexity while inducing stronger sparsity.

Abstract

This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g. sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of non-convex penalty functions (regularizers) constrained so as to ensure the convexity of the total cost function, F, to be minimized. The method is based on parametric penalty functions, the parameters of which are constrained to ensure convexity of F. It is shown that optimal parameters can be obtained by semidefinite programming (SDP). This maximally sparse convex (MSC) approach yields maximally non-convex sparsity-inducing penalty functions constrained such that the total cost function, F, is convex. It is demonstrated that iterative MSC (IMSC) can yield solutions substantially more sparse than the standard convex sparsity-inducing approach, i.e., L1 norm minimization.