Statistics Papers

Document Type

Journal Article

Date of this Version

7-1-2013

Publication Source

IEEE Transactions on Signal Processing

Volume

61

Issue

13

Start Page

3279

Last Page

3290

DOI

10.1109/TSP.2013.2259164

Abstract

This paper establishes new restricted isometry conditions for compressed sensing and affine rank minimization. It is shown for compressed sensing that δ<;i>kA<;/i>+θ<;i>k<;/i>,<;i>kA<;/i> <; 1 guarantees the exact recovery of all <;i>k<;/i> sparse signals in the noiseless case through the constrained <;i>l<;/i><;sub>1<;/sub> minimization. Furthermore, the upper bound 1 is sharp in the sense that for any ε > 0, the condition δ<;i>kA<;/i> + θ<;i>k<;/i>,<;i>kA<;/i> <; 1+ε is not sufficient to guarantee such exact recovery using any recovery method. Similarly, for affine rank minimization, if δ<;i>rM<;/i>+θ<;i>r<;/i>,<;i>rM<;/i> <; 1 then all matrices with rank at most <;i>r<;/i> can be reconstructed exactly in the noiseless case via the constrained nuclear norm minimization; and for any ε > 0, δ<;i>rM<;/i> +θ<;i>r<;/i>,<;i>rM<;/i> <; 1+ε does not ensure such exact recovery using any method. Moreover, in the noisy case the conditions δ<;i>kA<;/i>+θ<;i>k<;/i>,<;i>kA<;/i> <; 1 and δ<;i>rM<;/i>+θ<;i>r<;/i>,<;i>rM<;/i> <; 1 are also sufficient for the stable recovery of sparse signals and low-rank matrices respectively. Applications and extensions are also discussed.

Keywords

compressed sensing, matrix algebra, minimization, affine rank minimization, compressed sensing, constrained l1 minimization, constrained nuclear norm minimization, low-rank matrices, noiseless case, recovery method, restricted isometry conditions, sparse signals, compressed sensing, image reconstruction, noise measurement, signal processing, sparse matrices, vectors, Dantzig selector, compressed sensing, constrained nuclear norm minimization, low-rank matrix recovery, restricted isometry, sparse signal recovery

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Date Posted: 27 November 2017

This document has been peer reviewed.