Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery
Tony Cai and Anru Zhang
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Abstract:
This paper establishes a sharp condition on the restrict 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 δAk <1/3, then all k-sparse signals β can be recovered exactly via the constrained l1 minimization based on y = Aβ. Similarly, if the linear map M satisfies the RIP condition δMr <1/3, then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on b = 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.
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