Tensor principal component analysis via convex optimization

• 成果类型: Article
• 署名作者: Jiang, Bo; Ma, Shiqian; Zhang, Shuzhong
• 署名单位: Shanghai University of Finance & Economics; Chinese University of Hong Kong; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0774-0
• 发表日期: 2015
• 页码: 423-457
• 关键词: semidefinite; algorithms; rank; approximation

摘要

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.