A Third-order Generalization of the Matrix Svd as a Product of Third-order Tensors

Abstract. Traditionally, extending the Singular Value Decomposition (SVD) to third-order tensors (multiway arrays) has involved a representation using the outer product of vectors. These outer products can be written in terms of the n-mode product, which can also be used to describe a type of multiplication between two tensors. In this paper, we present a different type of third-order generalization of the SVD where an order-3 tensor is instead decomposed as a product of order-3 tensors. In order to define this new notion, we define tensor-tensor multiplication in such a way so that it is closed under this operation. This results in new definitions for tensors such as the tensor transpose, inverse, and identity. These definitions have the advantage they can be extended, though in a non-trivial way, to the order-p (p > 3) case [31]. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which could then be used in applications such as data compression. We therefore present two strategies for compressing thirdorder tensors which make use of our new SVD generalization and give some numerical comparisons to existing algorithms on synthetic data.