Asymptotic Rank Speedup Theorems, Revisited

TL;DR

This paper develops a unified framework for improving asymptotic tensor rank bounds, with applications to matrix multiplication complexity and tensor analysis.

Contribution

It generalizes classical speedup theorems, enabling quantitative improvements in asymptotic tensor rank bounds beyond previous results.

Findings

Asymptotic rank of small Coppersmith-Winograd tensor is less than its border rank.

Improved upper bound below d^{2ω/3} for asymptotic rank of any d×d×d tensor.

Introduces Strassen calculus for analyzing degenerations with nontrivial direct sums.

Abstract

Motivated by fast matrix multiplication and recent connections between asymptotic tensor rank and fine-grained complexity, we revisit classical tools from the matrix multiplication literature and develop a framework for obtaining improved asymptotic rank upper bounds for tensors beyond matrix multiplication. In the 1980s, Coppersmith-Winograd and Strassen discovered a series of speedup theorems for asymptotic rank: in certain regimes, one can extract additional terms from a border rank upper bound on a tensor $T$, and then use these terms to obtain an improved asymptotic rank of $T$. We establish general speedup theorems that subsume these results and enable quantitative improvements. Two representative applications are: (1) The asymptotic rank of the small Coppersmith-Winograd tensor $\mathrm{cw}_q$ is less than its border rank. For instance, we prove the asymptotic rank of $\mathrm{cw}_2$ is smaller than $3.931$, improving on $\underline{\mathrm{R}}(\mathrm{cw}_2)=4$. It is known that if the asymptotic rank of $\mathrm{cw}_2$ equals $3$, this would imply $\omega=2$. (2) A general improvement over Strassen's bound: we obtain an upper bound below $d^{2\omega/3}$ on the asymptotic rank of any $d\times d\times d$ tensor. To make full use of speedups, we analyze degenerations in which both sides are nontrivial direct sums, a setting where the optimal quantitative bound one can achieve was previously unclear. We do so via an approach we call Strassen calculus: a systematic method for converting such degeneration data into explicit asymptotic rank bounds using Strassen's theory of the asymptotic spectrum.