Tensor decomposition beyond uniqueness, with an application to the minrank problem

TL;DR

This paper extends the uniqueness theorem for tensor decompositions beyond the traditional setting, introduces an efficient randomized algorithm for minimum matrix-vector decompositions, and applies these results to compute minimum rank matrices in generic spaces.

Contribution

It generalizes Jennrich's uniqueness theorem using matrix-vector decomposition and provides an efficient algorithm for minimum rank tensor decomposition.

Findings

Proved a generalized uniqueness theorem for tensor decompositions.

Designed an efficient randomized algorithm for minimum matrix-vector decomposition.

Applied the results to compute all minimum rank matrices in certain vector spaces.

Abstract

We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank). As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices.