General Tensor Decomposition, Moment Matrices and Applications

TL;DR

This paper generalizes tensor decomposition beyond matrices, linking it to moment matrix problems and introducing a new algorithm for multihomogeneous tensors, with applications demonstrated through algebraic and eigenvector methods.

Contribution

It introduces a novel tensor decomposition method based on moment matrices and flat extension criteria, extending classical approaches to multihomogeneous tensors.

Findings

New criterion for flat extension of Quasi-Hankel matrices

Algorithm applicable to general multihomogeneous tensors

Decomposition recoverable via eigenvector computation

Abstract

The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.