Coherence-Based Robust Analysis of Basis Pursuit De-Noising and Beyond

Abstract

In this paper, by means of the powerful coherence tool, we first establish a coherence-based performance guarantee for the basis pursuit de-noising (BPDN) to recover the signals corrupted with the `2norm bounded noise, and then extend these theoretical results to guarantee the robust recovery of the signals in the presence of the Dantzig Selector (DS) type noise, as well as the structured block-sparse signal recovery in the presence of the bounded noise. To the best of our knowledge, we first extend nontrivially the (sharp) uniform recovery condition derived by Cai, Wang and Xu (2010) for the constrained l 1 -norm minimization model (which takes the form of μ <; 1/2k-1, where μ is defined as the coherence of A) to two unconstrained regularized l 1 -norm minimization models to guarantee the robust recovery of any signals (not necessary to be k-sparse) under the l 2 -norm bounded noise and the DS type noise settings, respectively. Moreover, a uniform recovery condition and its two resulting error estimates are also established for the first time to our knowledge, for the robust block-sparse signal recovery using a regularized mixed l 2 /l 1 -norm minimization model, and these results well complement the existing theoretical study on this model which focuses on the non-uniform recovery conditions and/or the robust signal recovery in presence of the random noise. Two real-world applications, i.e., the recovery of the grey images and the fetal electrocardiogram (FECG) signals, are conducted to support our claims.