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

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.