A Direct Method for Computing the Complex-Valued Triple Decomposition of Third-Order Tensors

Abstract. A direct method is first proposed to compute a complex-valued triple decomposition of a third-order tensor. The method operates under three assumptions: (i) The triple rank [Formula: see text] of an [Formula: see text] tensor satisfies [Formula: see text]. (ii) Two factor tensors of the triple decomposition are generic. (iii) The third factor tensor has linearly independent fibers. When [Formula: see text], the computational cost of the proposed direct method is approximately [Formula: see text] flops. Moreover, a sufficient condition for the essential uniqueness of the tensor triple decomposition is established under these assumptions. Numerical experiments demonstrate that our direct method is at least 10 times faster than alternating least squares and optimization-based iterative approaches. Finally, we showcase applications of the method in large-scale video analysis and stochastic partial differential equations using triple tensor representations.