Tensor principal component analysis via convex optimization

Bo Jiang, Shiqian Ma, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

This paper is concerned with the computation of the principal components for a general tensor, known as the tensor principal component analysis (PCA) problem. We show that the general tensor PCA problem is reducible to its special case where the tensor in question is super-symmetric with an even degree. In that case, the tensor can be embedded into a symmetric matrix. We prove that if the tensor is rank-one, then the embedded matrix must be rank-one too, and vice versa. The tensor PCA problem can thus be solved by means of matrix optimization under a rank-one constraint, for which we propose two solution methods: (1) imposing a nuclear norm penalty in the objective to enforce a low-rank solution; (2) relaxing the rank-one constraint by semidefinite programming. Interestingly, our experiments show that both methods can yield a rank-one solution for almost all the randomly generated instances, in which case solving the original tensor PCA problem to optimality. To further cope with the size of the resulting convex optimization models, we propose to use the alternating direction method of multipliers, which reduces significantly the computational efforts. Various extensions of the model are considered as well.

Original languageEnglish (US)
Pages (from-to)423-457
Number of pages35
JournalMathematical Programming
Volume150
Issue number2
DOIs
StatePublished - 2015

Keywords

  • Low rank
  • Nuclear norm
  • Principal component analysis
  • Semidefinite programming relaxation
  • Tensor

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