A Proof of Conjecture on Restricted Isometry Property Constants $\delta _{tk}\ \left(0<t<\frac {4}{3}\right)$

Abstract

In this paper, we give a complete answer to the conjecture on restricted isometry property (RIP) constants $\delta _{tk} (0 , which was proposed by T. Cai and A. Zhang. We have shown that when $0 , the condition $\delta _{tk} is sufficient to guarantee the exact recovery for all $k$ -sparse signals in the noiseless case via the constrained $\ell _{1}$ -norm minimization. These bounds are sharp in the sense that for any $\epsilon >0,\,\,\delta _{tk} cannot guarantee the exact recovery of some $k$ -sparse signals. Furthermore, it will be shown that similar characterizations also hold for low-rank matrix recovery. Thus, combined with T. Cai and A. Zhang’s work, a complete characterization for sharp RIP constants $\delta _{tk}$ for all $t > 0$ is obtained to guarantee the exact recovery of all $k$ -sparse signals and matrices with rank at most $k$ by $\ell _{1}$ -norm minimization and nuclear norm minimization, respectively. Noisy cases and approximately sparse cases are also considered. To solve the conjecture, we construct a few identities so that RIP of order $tk$ , which is the target of our main results, can be perfectly applied to them.