Date of this Version
IEEE Transactions on Information Theory
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.
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
Cai, T., & Zhang, A. (2014). Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices. IEEE Transactions on Information Theory, 60 (1), 122-132. http://dx.doi.org/10.1109/TIT.2013.2288639
Date Posted: 27 November 2017
This document has been peer reviewed.