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

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• 作者：T. Tony Cai ; ¹ ; ^{tcai@wharton.upenn.edu} ; Anru Zhang
• 关键词：Compressed sensing ; Dantzig selector ; ?1?1 minimization ; Low-rank matrix recovery ; Nuclear norm minimization ; Restricted isometry ; Sparse signal recovery
• 刊名：Applied and Computational Harmonic Analysis
• 出版年：2013
• 出版时间：July, 2013
• 年：2013
• 卷：35
• 期：1
• 页码：74-93
• 全文大小：342 K

文摘

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 , then all k-sparse signals ¦Â can be recovered exactly via the constrained minimization based on . Similarly, if the linear map satisfies the RIP condition , then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on . 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.