Statistics Papers

Document Type

Journal Article

Date of this Version

1-2014

Publication Source

IEEE Transactions on Information Theory

Volume

60

Issue

1

Start Page

122

Last Page

132

DOI

10.1109/TIT.2013.2288639

Abstract

This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrixrecovery. The analysis relies on a key technical tool, which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while yielding sharp results. It is shown that for any given constant t ≥ 4/3, in compressed sensing, δtkA <; √((t-1)/t) guarantees the exactrecovery of all k sparse signals in the noiseless case through the constrained l1 minimization, and similarly, in affine rank minimization, δtrM <; √((t-1)/t) ensures the exact reconstruction of all matriceswith rank at most r in the noiseless case via the constrained nuclear norm minimization. In addition, for any ε > 0, δtkA <; √(t-1/t) + ε is not sufficient to guarantee the exact recovery of all k-sparse signals for large k. Similar results also hold for matrix recovery. In addition, the conditions δtkA <; √((t-)1/t) and δtrM<; √((t-1)/t) are also shown to be sufficient, respectively, for stable recovery of approximately sparsesignals and low-rank matrices in the noisy case.

Keywords

compressed sensing, matrix algebra, minimisation, signal representation, affine rank minimization, compressed sensing, constrained l1 minimization, constrained nuclear norm minimization, k-sparse signal recovery, low-rank matrix recovery, sharp restricted isometry conditions, sparse polytope representation, sparse vectors, minimization methods, noise, noise measurement, sparse matrices, vectors, affine rank minimization, 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.