Date of this Version
Applied and Computational Harmonic Analysis
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 ℓ1 minimization based on y = Aβ. Similarly, if the linear map M satisfies the RIP condition δrM, 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.
© 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.
compressed sensing, dantzig selector, ℓ1 minimization, low-rank matrix recovery, nuclear norm minimization, restricted isometry, sparse signal recovery
Cai, T., & Zhang, A. (2013). Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery. Applied and Computational Harmonic Analysis, 35 (1), 74-93. http://dx.doi.org/10.1016/j.acha.2012.07.010
Date Posted: 27 November 2017
This document has been peer reviewed.