DocumentCode :
1451692
Title :
Recovering Low-Rank Matrices From Few Coefficients in Any Basis
Author :
Gross, David
Author_Institution :
Inst. for Theor. Phys., Leibniz Univ. Hannover, Hannover, Germany
Volume :
57
Issue :
3
fYear :
2011
fDate :
3/1/2011 12:00:00 AM
Firstpage :
1548
Lastpage :
1566
Abstract :
We present novel techniques for analyzing the problem of low-rank matrix recovery. The methods are both considerably simpler and more general than previous approaches. It is shown that an unknown matrix of rank can be efficiently reconstructed from only randomly sampled expansion coefficients with respect to any given matrix basis. The number quantifies the “degree of incoherence” between the unknown matrix and the basis. Existing work concentrated mostly on the problem of “matrix completion” where one aims to recover a low-rank matrix from randomly selected matrix elements. Our result covers this situation as a special case. The proof consists of a series of relatively elementary steps, which stands in contrast to the highly involved methods previously employed to obtain comparable results. In cases where bounds had been known before, our estimates are slightly tighter. We discuss operator bases which are incoherent to all low-rank matrices simultaneously. For these bases, we show that randomly sampled expansion coefficients suffice to recover any low-rank matrix with high probability. The latter bound is tight up to multiplicative constants.
Keywords :
computational complexity; matrix algebra; degree of incoherence; low-rank matrix recovery; matrix completion; multiplicative constants; randomly sampled expansion coefficients; Coherence; Compressed sensing; Convex functions; Eigenvalues and eigenfunctions; Linear matrix inequalities; Random variables; Compressed sensing; matrix completion; matrix recovery; operator large-deviation bound; quantum-state tomography;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.2011.2104999
Filename :
5714248
Link To Document :
بازگشت