Lower bounds on the rank and symmetric rank of real tensors

TL;DR

This paper develops new lower bounds on the rank and symmetric rank of real tensors using unfoldings and Sylvester's inequality, enabling the construction of tensors that disprove Comon's conjecture.

Contribution

It introduces a novel method to lower bound tensor ranks via unfoldings and constructs the first known real tensor counterexample to Comon's conjecture.

Findings

Established lower bounds on tensor rank and symmetric rank.

Constructed the first real tensor with differing rank and symmetric rank.

Validated a conjecture on symmetric rank lower bounds.

Abstract

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.