Analysis non-sparse recovery for relaxed ALASSO

Abstract

This paper considers recovery of signals that are corrupted with noise. We focus on a novel model which is called relaxed ALASSO (RALASSO) model introduced by Z. Tan et al. (2014). Compared to the well-known ALASSO, RALASSO can be solved better in practice. Z. Tan et al. (2014) used the $ D $-RIP to characterize the sparse or approximately sparse solutions for RALASSO when the $ D $-RIP constant $ \delta_{2k} < 0.1907 $, where the solution is sparse or approximately sparse in terms of a tight frame $ D $. However, their estimate of error bound for solution heavily depends on the term $ \Vert D^*D\Vert_{1, 1} $. Besides, compared to other works on signals recovering from ALASSO, the condition $ \delta_{2k} < 0.1907 $ is even stronger. Based on the RALASSO model, we use new methods to get a better estimate of error bound and give a weaker sufficient condition in this article for the inadequacies of the results by Z. Tan et al. (2014). One of the result of this paper is to use another method called the robust $ \ell_2 $ $ D $-Null Space Property to obtain the sparse or non-sparse solution of RALASSO and give the error estimation of RALASSO, where we eliminate the term $ \Vert D^*D\Vert_{1, 1} $ in the constants. Another result of the paper is to utilize the $ D $-RIP to obtain a new condition $ \delta_{2k} < 0.3162 $ which is weaker than the condition $ \delta_{2k} < 0.1907 $. To some extent, RALASSO is equivalent to ALASSO and the condition is also weaker than the similar one $ \delta_{3k} < 0.25 $ by J. Lin, and S. Li (2014) and $ \delta_{2k}<0.25 $ by Y. Xia, and S. Li (2016).