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Proceedings Paper

Uniqueness conditions for low-rank matrix recovery
Author(s): Y. C. Eldar; D. Needell; Y. Plan
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Paper Abstract

Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractable approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m ≥ Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever - tractable or not. We show that for a family of random measurement ensembles, m ≥ 4nr-4r2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r matrices by rank minimization. Furthermore, this value of m precisely matches the dimension of the manifold of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m ≥ 2nr - r2 + 1 random measurements are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may compare the efficacy of nuclear-norm minimization.

Paper Details

Date Published: 14 September 2011
PDF: 9 pages
Proc. SPIE 8138, Wavelets and Sparsity XIV, 81380M (14 September 2011); doi: 10.1117/12.891933
Show Author Affiliations
Y. C. Eldar, Technion-Israel Institute of Technology (Israel)
D. Needell, Stanford Univ. (United States)
Y. Plan, California Institute of Technology (United States)


Published in SPIE Proceedings Vol. 8138:
Wavelets and Sparsity XIV
Manos Papadakis; Dimitri Van De Ville; Vivek K. Goyal, Editor(s)

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