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Scalable Low-Rank Representation

  • Guangcan Liu
  • Shuicheng Yan
Chapter

Abstract

While the optimization problem associated with LRR is convex and easy to solve, it is actually a big challenge to achieve high efficiency, especially under large-scale settings. In this chapter we therefore address the problem of solving nuclear norm regularized optimization problems (NNROPs), which contain a category of problems including LRR. Based on the fact that the optimal solution matrix to an NNROP is often low-rank, we revisit the classic mechanism of low-rank matrix factorization, based on which we present an active subspace algorithm for efficiently solving NNROPs by transforming large-scale NNROPs into small-scale problems. The transformation is achieved by factorizing the large-size solution matrix into the product of a small-size orthonormal matrix (active subspace) and another small-size matrix. Although such a transformation generally leads to non-convex problems, we show that suboptimal solution can be found by the augmented Lagrange alternating direction method. For the robust PCA (RPCA) [7] problem, which is a typical example of NNROPs, theoretical results verify sub-optimality of the solution produced by our algorithm. For the general NNROPs, we empirically show that our algorithm significantly reduces the computational complexity without loss of optimality.

Keywords

Nuclear norm optimization Active subspace Matrix factorization Stiefel manifold Alternating direction method 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Cornell UniversityIthacaUSA
  2. 2.National University of SingaporeKent RidgeSingapore

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