Mathematical Programming Computation

, Volume 4, Issue 4, pp 333–361 | Cite as

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

Full Length Paper

Abstract

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.

Keywords

Matrix completion Alternating minimization Nonlinear GS method Nonlinear SOR method 

Mathematics Subject Classification (2000)

65K05 90C06 93C41 68Q32 

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Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Institute of Natural SciencesShanghai Jiaotong UniversityShanghaiChina
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

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