Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm

The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions—a task that is increasingly costly as matrix sizes and ranks incr...

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Veröffentlicht in:Mathematical programming computation Jg. 4; H. 4; S. 333 - 361
Hauptverfasser: Wen, Zaiwen, Yin, Wotao, Zhang, Yin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer-Verlag 01.12.2012
Schlagworte:
Full Length Paper
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Theory of Computation
65K05
90C06
Alternating minimization
Nonlinear GS method
93C41
68Q32
Nonlinear SOR method
Matrix completion
ISSN:1867-2949, 1867-2957
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Beschreibung
Zusammenfassung:The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions—a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale problems, we propose a low-rank factorization model and construct a nonlinear successive over-relaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Extensive numerical experiments show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclear-norm minimization algorithms. In addition, convergence of this nonlinear SOR algorithm to a stationary point is analyzed.
ISSN:1867-2949
1867-2957
DOI:10.1007/s12532-012-0044-1