Iterative $\ell_1$ minimization for non-convex compressed sensing

TL;DR

This paper introduces an iterative $ ext{l}_1$ minimization algorithm based on DCA for non-convex compressed sensing, demonstrating superior sparse recovery performance over traditional methods, especially in ill-conditioned scenarios and MRI applications.

Contribution

The paper proposes a novel iterative $ ext{l}_1$ minimization framework that guarantees improved sparse recovery and robustness over basis pursuit, with significant empirical advantages.

Findings

Outperforms basis pursuit in success rates of sparse recovery.

Excels over state-of-the-art $ ext{l}_{1/2}$ and logarithmic minimizations in ill-conditioned regimes.

Successfully recovers MRI images with minimal projections.

Abstract

An algorithmic framework, based on the difference of convex functions algorithm (DCA), is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence of $\ell_1$ minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit ($\ell_1$ minimization) and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literature, our method always out-performs basis pursuit, no matter how ill-conditioned the measurement matrix is. Moreover, the iterative $\ell_1$ (IL$_1$) algorithm lead by a wide margin the state-of-the-art algorithms on $\ell_{1/2}$ and logarithimic minimizations in the strongly coherent (highly ill-conditioned) regime, despite the same objective functions. Last but not least, in the application of magnetic resonance imaging (MRI), IL$_1$ algorithm easily recovers the phantom image with just 7 line projections.