High-Order CPD Estimation with Dimensionality Reduction Using a Tensor Train Model

Abstract

The canonical polyadic decomposition (CPD) is one of the most popular tensor-based analysis tools due to its usefulness in numerous fields of application. The Q-order CPD is parametrized by $Q$ matrices also called factors which have to be recovered. The factors estimation is usually carried out by means of the alternating least squares (ALS) algorithm. In the context of multi-modal big data analysis, i.e., large order $(Q)$ and dimensions, the ALS algorithm has two main drawbacks. Firstly, its convergence is generally slow and may fail, in particular for large values of $Q$, and secondly it is highly time consuming. In this paper, it is proved that a Q-order CPD of rank-R is equivalent to a train of $Q$ 3-order CPD(s) of rank-R. In other words, each tensor train (TT)-core admits a 3-order CPD of rank-R. Based on the structure of the TT-cores, a new dimensionality reduction and factor retrieval scheme is derived. The proposed method has a better robustness to noise with a smaller computational cost than the ALS algorithm.