From computation to comparison of tensor decompositions

TL;DR

This paper presents a method to verify if two tensors are generated by the same terms without computing their explicit decompositions, using linear algebra techniques for efficient and reliable comparison.

Contribution

It introduces a novel approach to compare tensor decompositions via column and row space analysis, reducing complexity and improving numerical stability.

Findings

Verification reduces to linear algebra comparison

Number of terms and ranks can be inferred

Applicable to multi-label classification in blind source separation

Abstract

Decompositions of higher-order tensors into sums of simple terms are ubiquitous. We show that in order to verify that two tensors are generated by the same (possibly scaled) terms it is not necessary to compute the individual decompositions. In general the explicit computation of such a decomposition may have high complexity and can be ill-conditioned. We now show that under some assumptions the verification can be reduced to a comparison of both the column and row spaces of the corresponding matrix representations of the tensors. We consider rank-1 terms as well as low multilinear rank terms (also known as block terms) and show that the number of the terms and their multilinear rank can be inferred as well. The comparison relies only on numerical linear algebra and can be done in a numerically reliable way. We also illustrate how our results can be applied to solve a multi-label classification problem that appears in the context of blind source separation.