Successive Concave Sparsity Approximation for Compressed Sensing

TL;DR

This paper introduces a novel iterative algorithm for sparse signal recovery in compressed sensing, utilizing successively accurate concave approximations of the $ ext{l}_0$ norm, with proven convergence and superior performance in simulations.

Contribution

It proposes a new successively approximation-based algorithm for sparse recovery, with theoretical guarantees and improved empirical performance over existing methods.

Findings

Algorithm closely follows oracle estimator performance

Provides convergence guarantees for iterative thresholding

Outperforms state-of-the-art algorithms in simulations

Abstract

In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell_1$ and $\ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted $\ell_1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.