Alternating direction algorithms for $\ell_0$ regularization in compressed sensing

TL;DR

This paper introduces three new greedy algorithms for compressed sensing based on ADMM, providing theoretical guarantees and outperforming existing methods in reconstructing specific signal types.

Contribution

The paper proposes three novel iterative greedy algorithms derived from ADMM for $ ext{l}_0$-regularized least squares, with theoretical guarantees and improved performance over existing methods.

Findings

Algorithms outperform IHT, NIHT, and HTP in experiments.

First theoretical guarantee of ADMM for $ ext{l}_0$-LS under RIP.

Effective in reconstructing CARS and Gaussian signals.

Abstract

In this paper we propose three iterative greedy algorithms for compressed sensing, called \emph{iterative alternating direction} (IAD), \emph{normalized iterative alternating direction} (NIAD) and \emph{alternating direction pursuit} (ADP), which stem from the iteration steps of alternating direction method of multiplier (ADMM) for $\ell_0$-regularized least squares ($\ell_0$-LS) and can be considered as the alternating direction versions of the well-known iterative hard thresholding (IHT), normalized iterative hard thresholding (NIHT) and hard thresholding pursuit (HTP) respectively. Firstly, relative to the general iteration steps of ADMM, the proposed algorithms have no splitting or dual variables in iterations and thus the dependence of the current approximation on past iterations is direct. Secondly, provable theoretical guarantees are provided in terms of restricted isometry property, which is the first theoretical guarantee of ADMM for $\ell_0$-LS to the best of our knowledge. Finally, they outperform the corresponding IHT, NIHT and HTP greatly when reconstructing both constant amplitude signals with random signs (CARS signals) and Gaussian signals.