Asymptotic rank bounds: a numerical census

TL;DR

This paper systematically computes improved asymptotic rank bounds for tensors using numerical methods, providing data that supports Strassen's conjecture and highlighting computational challenges.

Contribution

It introduces a numerical approach to improve asymptotic tensor rank bounds and applies it across feasible cases, advancing understanding of tensor complexity.

Findings

New asymptotic rank bounds that surpass generic border rank bounds

Numerical evidence supporting Strassen's asymptotic rank conjecture

Identification of computational barriers in current numerical methods

Abstract

We systematically compute improved asymptotic rank bounds for tensors. Using numerical implicitization, we implement the geometric framework of Kaski and Micha{\l}ek across all computationally feasible cases. By detecting the absence of low-degree vanishing polynomials on secant varieties, we obtain new asymptotic rank bounds that improve upon the generic border rank bounds. The results provide numerical data supporting Strassen's asymptotic rank conjecture and clarify the computational barriers posed by current numerical methods.