Hankel tensor decompositions and ranks

TL;DR

This paper explores the ranks of Hankel tensors, providing algorithms for Vandermonde rank computation and proving the equivalence of various ranks for generic tensors of certain orders, confirming a conjecture in these cases.

Contribution

It introduces an algorithm for Vandermonde rank computation and proves the equivalence of multiple tensor ranks for generic Hankel tensors of specific orders.

Findings

Vandermonde ranks can be computed algorithmically.

Various tensor ranks coincide for generic Hankel tensors of even order or order three.

Supports the validity of Comon's conjecture in these cases.

Abstract

Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional Hankel tensor of even order or order three, we prove that the the cp rank, symmetric rank, border rank, symmetric border rank, and Vandermonde rank all coincide with each other. In particular, this implies that the Comon's conjecture is true for generic Hankel tensors when the order is even or three. Some open questions are also posed.