Typical ranks of random order-three tensors

TL;DR

This paper investigates the typical ranks of real three-dimensional tensors, providing geometric proofs, probabilistic interpretations, and linking tensor rank probabilities to the geometry of real cubic surfaces.

Contribution

It offers a geometric proof for the typical ranks of certain tensors and connects rank probabilities to intersection points of linear spaces with Segre varieties.

Findings

Typical ranks of certain tensors are contained in {ℓ, ℓ+1}

Probabilities of ranks relate to intersection points with Segre varieties

Bound on expected number of real lines on a cubic surface

Abstract

In this paper we study typical ranks of real $m\times n \times \ell$ tensors. In the case $ (m-1)(n-1)+1 \leq \ell \leq mn$ the typical ranks are contained in $\{\ell, \ell +1\}$, and $\ell$ is always a typical rank. We provide a geometric proof of this fact. We express the probabilities of these ranks in terms of the probabilities of the numbers of intersection points of a random linear space with the Segre variety. In addition, we give some heuristics to understand how the probabilities of these ranks behave, based on asymptotic results on the average number of real points in a random linear slice of a Segre variety with a subspace of complementary dimension. The typical ranks of real $3\times 3\times 5$ tensors are $5$ and $6$. We link the rank probabilities of a $3\times 3 \times 5$ tensor with i.i.d.\ Gaussian entries to the probability of a random cubic surface in $\P^3$ having real lines. As a consequence, we get a bound on the expected number of real lines on such a surface.