Statistics Papers

Document Type

Journal Article

Date of this Version

7-2013

Publication Source

Applied and Computational Harmonic Analysis

Volume

35

Issue

1

Start Page

74

Last Page

93

DOI

10.1016/j.acha.2012.07.010

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 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.

Copyright/Permission Statement

© 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.

Keywords

compressed sensing, dantzig selector, ℓ1 minimization, low-rank matrix recovery, nuclear norm minimization, restricted isometry, sparse signal recovery

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Date Posted: 27 November 2017

This document has been peer reviewed.