Tight Lower Bound on the Tensor Rank based on the Maximally Square Unfolding

Abstract

Tensors decompositions are a class of tools for analysing datasets of high dimensionality and variety in a natural manner, with the Canonical Polyadic Decomposition (CPD) being 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 address this issue, we investigate tight lower bounds on $R$ with the aim to provide a reduced search space, and hence to lessen the computational costs of the 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 with aspect ratio closest to unity (maximally square). Moreover, we demonstrate that, for a generic tensor, such lower bound can be attained under very mild conditions, whereby the tensor rank becomes detectable. Numerical examples demonstrate the benefits of this result.