A Proof of Conjecture on Restricted Isometry Property Constants $\delta _{tk}\ \left(0

In this paper, we give a complete answer to the conjecture on restricted isometry property (RIP) constants δ_tk(0<;t<;(4/3)), which was proposed by T. Cai and A. Zhang. We have shown that when 0 <; t <; (4/3), the condition δ_tk <; (t/(4-t)) is sufficient to guarantee the exact recovery for all k-sparse signals in the noiseless case via the constrained ℓ₁-norm minimization. These bounds are sharp in the sense that for any ϵ>0, δ_tk <; (t/(4-t)) + ϵ 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 δ_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 ℓ₁-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.