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TR20084
A Thirdorder Generalization of the Matrix SVD as a Product of Thirdorder Tensors 

Authors:  Kilmer, Misha E.; Martin, Carla D.; Perrone, Lisa 
Date:  20081004 
Pages:  20 
Download Formats:  [PDF] 
Traditionally, extending the Singular Value Decomposition (SVD) to thirdorder tensors (multiway arrays) has involved a representation using the outer product of vectors. These outer products can be written in terms of the nmode product, which can also be used to describe a type of multiplication between two tensors. In this paper, we present a different type of thirdorder generalization of the SVD where an order3 tensor is instead decomposed as a product of order3 tensors. In order to define this new notion, we define tensortensor 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 nontrivial way, to the orderp (p > 3) case. 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. 
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