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

TL;DR

This paper establishes a tight lower bound on tensor rank using the maximally square unfolding, reducing computational complexity in tensor decomposition by linking rank bounds to tensor dimensions.

Contribution

It introduces a novel lower bound on tensor rank based on the maximally square unfolding, enabling more efficient tensor rank detection for generic tensors.

Findings

Lower bounds can be attained under mild conditions.

The approach reduces the search space for tensor rank evaluation.

Numerical examples confirm the effectiveness of the bounds.

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.