Non-orthogonal tensor diagonalization

Petr Tichavský, Anh Huy Phan, Andrzej Cichocki

Abstract: Highlights • A novel non-orthogonal tensor diagonalization method is proposed. • It can have form of two sided or three sided diagonalization, or block-diagonalization. • It can serve as a new method of Canonic Polyadic (CP) tensor decomposition, but even more importantly for block-term tensor decomposition. • The proposed method has low computational complexity, similar to that of the popular Alternating Least Squares algorithm for CP decomposition. • In comparison with other CP decomposition and block-term decomposition methods, the proposed one is less sensitive to wrong initialization. Abstract Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It has a link to an approximate joint diagonalization (AJD) of a set of matrices. In this paper, we derive (1) a new algorithm for a symmetric AJD, which is called two-sided symmetric diagonalization of an order-three tensor, (2) a similar algorithm for a non-symmetric AJD, also called a two-sided diagonalization of an order-three tensor, and (3) an algorithm for three-sided diagonalization of order-three or order-four tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and in certain scenarios they can outperform traditional CP decomposition methods. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to tensor block-term decomposition. The proposed algorithm can either outperform the existing block-term decomposition algorithms, or produce good initial points for their application.