New RIC Bounds via l_q-minimization with 0<q<=1 in Compressed Sensing

TL;DR

This paper establishes new bounds on restricted isometry constants for l_q-minimization (0<q<=1) in compressed sensing, improving conditions for exact sparse signal recovery.

Contribution

It introduces novel RIC bounds for l_q relaxations, including specific bounds for q=1/2 and q=1, enhancing recovery guarantees over previous results.

Findings

Exact recovery via l_{1/2} and l_1 minimizations under certain RIC bounds.

Derived sufficient conditions for RIC bounds depending on q and sparsity.

Provided explicit RIC bounds for various q values, including q=1/2 and q=1.

Abstract

The restricted isometry constants (RICs) play an important role in exact recovery theory of sparse signals via l_q(0<q<=1) relaxations in compressed sensing. Recently, Cai and Zhang[6] have achieved a sharp bound \delta_tk<\sqrt{1-1/t} for t>=4/3 to guarantee the exact recovery of k sparse signals through the l_1 minimization. This paper aims to establish new RICs bounds via l_q(0<q<=1) relaxation. Based on a key inequality on l_q norm, we show that (i) the exact recovery can be succeeded via l_{1/2} and l_1 minimizations if \delta_tk<\sqrt{1-1/t} for any t>1, (ii)several sufficient conditions can be derived, such as for any 0<q<1/2, \delta_2k<0.5547 when k>=2, for any 1/2<q<1, \delta_2k<0.6782 when k>=1, (iii) the bound on \delta_k is given as well for any 0<q<=1, especially for q=1/2,1, we obtain \delta_k<1/3 when k(>=2) is even or \delta_k<0.3203 when k(>=3) is odd.