A lower bound on the tensor rank based on its maximally square matrix unfolding

Abstract

Tensor decompositions represent a class of tools for analysing datasets of high dimensionality and variety in a natural manner, with the Canonical Polyadic Decomposition (CPD) serving as a main pillar. While the notion of CPD is closely intertwined with that of the tensor rank, R, unlike the matrix rank, the computation of the tensor rank is an NP-hard problem, owing to the associated computational burden of evaluating the CPD. To help alleviate this issue, we investigate lower bounds on the tensor rank, with the aim to provide a reduced search space, and hence relax the computational costs of CPD evaluation. This is achieved by establishing a link between the maximum attainable lower bound on R and the dimensions of the matrix unfolding of the tensor for which the aspect ratio is closest to unity (maximally square). Moreover, we demonstrate that such lower bound can be attained under very mild conditions, which facilitates the efficient identification and computation of the actual tensor rank. Numerical examples and a real-world application demonstrate the benefits of this approach.