Sharp RIP bound for sparse signal and low-rank matrix recovery

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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 δkA<1/3, then all k-sparse signals β can be recovered exactly via the constrained 1 minimization based on y=Aβ. Similarly, if the linear map M satisfies the RIP condition δrM<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.

Keywords

Compressed sensing
Dantzig selector
1 minimization
Low-rank matrix recovery
Nuclear norm minimization
Restricted isometry
Sparse signal recovery

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The research of Tony Cai was supported in part by NSF FRG Grant DMS-0854973.