Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery

TL;DR

This paper proves a precise RIP threshold of 1/3 for exact recovery of sparse signals and low-rank matrices, establishing the limits of success and failure in these recovery problems.

Contribution

It introduces a sharp RIP bound of 1/3 for exact recovery in sparse and low-rank matrix problems, clarifying the precise conditions needed for success.

Findings

Exact recovery guaranteed if RIP constant < 1/3

Recovery impossible if RIP constant ≥ 1/3

Provides oracle inequalities in noisy scenarios

Abstract

This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix $A$ satisfies the RIP condition $\delta_k^A<1/3$, then all $k$-sparse signals $\beta$ can be recovered exactly via the constrained $\ell_1$ minimization based on $y=A\beta$. Similarly, if the linear map $\cal M$ satisfies the RIP condition $\delta_r^{\cal M}<1/3$, then all matrices $X$ of rank at most $r$ can be recovered exactly via the constrained nuclear norm minimization based on $b={\cal M}(X)$. Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.