Stable Cosparse Recovery via \ell_p-analysis Optimization

TL;DR

This paper investigates $\, ext{ extlbrackdbl}p ext{ extrbrackdbl}$-analysis optimization for cosparse signal recovery, establishing error bounds, demonstrating advantages of nonconvex approaches, and proposing an iterative reweighted algorithm with promising empirical results.

Contribution

It introduces a nonconvex $\, ext{ extlbrackdbl}p ext{ extrbrackdbl}$-analysis optimization method for cosparse recovery, with theoretical error bounds and an iterative algorithm outperforming convex methods.

Findings

Nonconvex $\, ext{ extlbrackdbl}p ext{ extrbrackdbl}$-analysis outperforms convex methods in recovery accuracy.

Error bounds established via restricted $p$-isometry property.

Empirical results show the proposed method's superior performance.

Abstract

In this paper we study the $\ell_p$-analysis optimization ($0<p\leq1$) problem for cosparse signal recovery. We establish a bound for recovery error via the restricted $p$-isometry property over any subspace. We further prove that the nonconvex $\ell_q$-analysis optimization can do recovery with a lower sample complexity and in a wider range of cosparsity than its convex counterpart. In addition, we develop an iteratively reweighted method to solve the optimization problem under a variational framework. Empirical results of preliminary computational experiments illustrate that the nonconvex method outperforms its convex counterpart.