Iterative ${\ell }_1$ minimization for non-convex compressed sensing

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.

References in corpus (2)

• Nonconvex Sorted ${\ell }_1$ Minimization for Sparse Approximation
• Minimization of Transformed $L_1$ Penalty: Theory, Difference of Convex Function Algorithm, and Robust Application in Compressed Sensing