NDLI: Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices

Content Provider	IEEE Xplore Digital Library
Author	Cai, T.T. Anru Zhang
Copyright Year	1963
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 matrix recovery. 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, δ_tk^A <; √((t-1)/t) guarantees the exact recovery of all k sparse signals in the noiseless case through the constrained l₁ minimization, and similarly, in affine rank minimization, δ_tr^M <; √((t-1)/t) ensures the exact reconstruction of all matrices with rank at most r in the noiseless case via the constrained nuclear norm minimization. In addition, for any ε > 0, δ_tk^A <; √(^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 δ_tk^A <; √((t-)1/t) and δ_tr^M <; √((t-1)/t) are also shown to be sufficient, respectively, for stable recovery of approximately sparse signals and low-rank matrices in the noisy case.
Sponsorship	IEEE Information Theory Society
Starting Page	122
Ending Page	132
Page Count	11
File Size	349345
File Format	PDF
ISSN	00189448
Volume Number	60
Issue Number	1
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-01-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Sparse matrices Vectors Compressed sensing Noise measurement Noise Minimization methods sparse signal recovery Affine rank minimization compressed sensing constrained $\ell_{1}$ minimization low-rank matrix recovery constrained nuclear norm minimization restricted isometry
Content Type	Text
Resource Type	Article
Subject	Library and Information Sciences Information Systems Computer Science Applications

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