Baitian Li

Josh Alman, Baitian Li, Kevin Pratt

Strassen founded the theory of the asymptotic spectrum of tensors to study the complexity of matrix multiplication. A central challenge in this theory is to explicitly construct new spectral points. In Crelle 1991, Strassen proposed the upper support functionals $ζ^θ$ as candidate spectral points, where $θ$ ranges over a triangle $Θ$. Recent progress, involving tools and ideas from quantum information theory (Christandl-Vrana-Zuiddam, STOC 2018, JAMS 2021) and convex optimization (Hirai, 2025), culminated in the proof that the upper support functionals are indeed spectral points over the complex numbers (Sakabe-Doğan-Walter, 2026). In this paper, we give an even clearer picture of the situation for support functionals when $θ$ lies along the edges of the triangle. We show that not only are these functionals spectral points, but that they are uniquely determined as spectral points by their behavior on matrix multiplication tensors. As our methods are algebraic, as a corollary this establishes for the first time the existence of nontrivial spectral points over arbitrary fields. As part of our argument, we show a close connection between the edge support functionals and Harder-Narasimhan filtrations from quiver representation theory. We thus show, using recent work in algorithmic invariant theory, that these support functionals can be computed in deterministic polynomial time. Other ingredients of our proof include a new criterion for abstractly characterizing asymptotic tensor ranks by spectral points, and a characterization of the edge support functionals in terms of matrix multiplication capacity. As another application of these tools, we prove the existence of spectral points for higher-mode tensors beyond those currently known.

Josh Alman, Baitian Li

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 $ω=2$. (2) A general improvement over Strassen's bound: we obtain an upper bound below $d^{2ω/3}$ 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.

Baitian Li

We show that the hafnian of a symmetric $2n\times 2n$ matrix of $\operatorname{poly}(n)$-bit integers (which counts the number of perfect matchings of a $2n$-vertex graph) and the number of Hamiltonian cycles of an $n$-vertex directed graph can be computed in time $2^{n-Ω(\sqrt{n})}$, improving and generalizing an earlier algorithm of Björklund, Kaski, and Williams (Algorithmica 2019) that runs in time $2^{n - Ω\left(\sqrt{n/\log \log n}\right)}$. A key tool of our approach is the design of a data structure that supports fast evaluation of high-order derivatives of hafnian and Hamiltonian cycles, which integrates with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh, Guo, Kumar, and Umans (FOCS 2022, JACM 2024).