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Uniqueness conditions for low-rank matrix recovery

Eldar, Y. C. and Needell, D. and Plan, Y. (2011) Uniqueness conditions for low-rank matrix recovery. In: Wavelets and Sparsity XIV. Proceedings of SPIE. No.8138. Society of Photo-Optical Instrumentation Engineers , Bellingham, WA, Art. No. 81380M. ISBN 978-0-81948-748-3.

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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-4r^2 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 – r^2 + 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.

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Additional Information:© 2011 SPIE. We would like to thank Boris Bertman and Rohit Thomas for thoughtful discussions. This work was partially supported by the NSF DMS EMSW21-VIGRE grant.
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Subject Keywords:rank-minimization, nuclear norm minimization, low-rank matrix recovery, random matrices, compressed sensing
Series Name:Proceedings of SPIE
Issue or Number:8138
Record Number:CaltechAUTHORS:20161028-132335892
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Official Citation:Y. C. Eldar ; D. Needell ; Y. Plan; Uniqueness conditions for low-rank matrix recovery. Proc. SPIE 8138, Wavelets and Sparsity XIV, 81380M (September 13, 2011); doi:10.1117/12.891933
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71581
Deposited By: Tony Diaz
Deposited On:28 Oct 2016 20:32
Last Modified:03 Oct 2019 16:08

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