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

• Published in: IEEE transactions on information theory Vol. 60; no. 1; pp. 122 - 132
• Main Authors: Cai, T. Tony, Anru Zhang
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-01-2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Affine rank minimization; Applied sciences; Compressed sensing; constrained ell_{1} minimization; constrained nuclear norm minimization; Convex analysis; Detection, estimation, filtering, equalization, prediction; Electrical engineering; Exact sciences and technology; Information theory; Information, signal and communications theory; low-rank matrix recovery; Minimization methods; Noise; Noise measurement; restricted isometry; Sampling, quantization; Signal and communications theory; Signal, noise; Sparse matrices; sparse signal recovery; Telecommunications and information theory; Vectors; Polytope; low-rank matrix recovery; Isometry; restricted isometry; Affine rank minimization; constrained ℓ; Constrained optimization; sparse signal recovery; Signal restoration; minimization; Signal processing; Sparse representation; Signal reconstruction; Compressed sensing; constrained nuclear norm minimization
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2288639

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 l 1 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.