Enhanced Sparsity by Non-Separable Regularization

TL;DR

This paper introduces a novel non-separable non-convex penalty function that enhances sparsity in one-dimensional deconvolution, enabling convex formulation of ill-conditioned inverse problems and improving over traditional L1 regularization.

Contribution

It proposes a new non-separable non-convex penalty function that ensures convexity in sparse deconvolution, overcoming limitations of separable regularization methods.

Findings

The new penalty improves sparsity recovery in deconvolution tasks.

Explicit conditions for penalty parameter selection to maintain convexity.

An efficient forward-backward splitting algorithm for sparse deconvolution.

Abstract

This paper develops a convex approach for sparse one-dimensional deconvolution that improves upon L1-norm regularization, the standard convex approach. We propose a sparsity-inducing non-separable non-convex bivariate penalty function for this purpose. It is designed to enable the convex formulation of ill-conditioned linear inverse problems with quadratic data fidelity terms. The new penalty overcomes limitations of separable regularization. We show how the penalty parameters should be set to ensure that the objective function is convex, and provide an explicit condition to verify the optimality of a prospective solution. We present an algorithm (an instance of forward-backward splitting) for sparse deconvolution using the new penalty.