A class of null space conditions for sparse recovery via nonconvex, non-separable minimizations

TL;DR

This paper develops new null space conditions that guarantee sparse recovery using nonconvex, non-separable regularizations, extending theoretical understanding beyond traditional separable cases and potentially improving recovery performance.

Contribution

It introduces general null space conditions for nonconvex, non-separable regularizations, broadening the theoretical framework for sparse recovery guarantees.

Findings

Null space conditions for nonconvex, non-separable regularizations established

Conditions are less demanding than standard null space property for  minimization

Provides theoretical guarantees for exact, uniform recovery with nonconvex penalties

Abstract

For the problem of sparse recovery, it is widely accepted that nonconvex minimizations are better than $\ell_1$ penalty in enhancing the sparsity of solution. However, to date, the theory verifying that nonconvex penalties outperform (or are at least as good as) $\ell_1$ minimization in exact, uniform recovery has mostly been limited to separable cases. In this paper, we establish general recovery guarantees through null space conditions for nonconvex, non-separable regularizations, which are slightly less demanding than the standard null space property for $\ell_1$ minimization.