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Tensor Principal Component Analysis via Convex Optimization (2012)
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BibTeX
@MISC{Jiang12tensorprincipal,
author = {Bo Jiang and Shiqian Ma and Shuzhong Zhang and Hong Kong},
title = {Tensor Principal Component Analysis via Convex Optimization},
year = {2012}
}
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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 supersymmetric 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 rankone 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 yield a rank-one solution with high probability, thereby solving the original tensor PCA problem to optimality with high probability. 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.
Keyphrases
tensor principal component analysis convex optimization high probability rank-one constraint tensor pca problem computational effort vice versa symmetric matrix rank-one solution special case convex optimization model direction method general tensor pca problem original tensor pca problem various extension embedded matrix principal component general tensor semidefinite programming nuclear norm penalty solution method matrix optimization low-rank solution
