Abstract

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 $\ell 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.

Original languageEnglish (US)
Article number6121985
Pages (from-to)237-247
Number of pages11
JournalIEEE Transactions on Information Theory
Volume58
Issue number1
DOIs
StatePublished - Jan 2012

Keywords

  • Gradient search
  • Grassmann manifold
  • low-rank matrix completion
  • nonconvex optimization
  • performance guarantee

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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