Tensor Principal Component Analysis via Convex Optimization

TL;DR

This paper introduces a convex optimization approach for tensor PCA by reducing the problem to symmetric matrices, employing nuclear norm penalties and semidefinite programming, and demonstrates high-probability success with efficient algorithms.

Contribution

It presents a novel reduction of tensor PCA to matrix optimization and proposes effective convex relaxation methods with scalable algorithms.

Findings

High probability of obtaining rank-one solutions

Effective reduction of tensor PCA to matrix optimization

Use of ADMM for computational efficiency

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 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.