Bounds on strongly orthogonal ranks of tensors

Abstract

A strongly orthogonal decomposition of a tensor is a rank-one tensor decomposition with the two component vectors in each mode of any two rank-one tensors are either colinear or orthogonal. A strongly orthogonal decomposition with the smallest possible number of rank-one tensors is a strongly orthogonal rank decomposition. Any tensor has a strongly orthogonal rank decomposition. The number of rank-one tensors in a strongly orthogonal rank decomposition is the strongly orthogonal rank. In this article, bounds on the strongly orthogonal rank of a real tensor are investigated. A universal upper bound, in terms of the multilinear ranks, for the strongly orthogonal ranks is given for an arbitrary tensor space. A formula for the expected strongly orthogonal rank of a given tensor space is also given, which is verified for a set of tensor spaces numerically.