期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 58, 期 1, 页码 237-247出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2011.2171521
关键词
Gradient search; Grassmann manifold; low-rank matrix completion; nonconvex optimization; performance guarantee
资金
- Direct For Computer & Info Scie & Enginr
- Division of Computing and Communication Foundations [1117980] Funding Source: National Science Foundation
The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion have been proposed so far, it remains an open problem to devise l(0)-type search procedures with provable performance guarantees. The standard approach to the problem, which involves the minimization of an objective function defined using the Frobenius metric, has inherent difficulties: the objective function is not continuous and the solution set is not closed. To address this problem, we consider an optimization procedure that searches for a column (or row) space that is geometrically consistent with the partial observations. The geometric objective function is continuous everywhere and the solution set is the closure of the solution set of the Frobenius metric. We also preclude the existence of local minimizers, and hence establish strong performance guarantees, for special completion scenarios, which do not require matrix incoherence and hold with probability one for arbitrary matrix size.
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