# Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

arXiv: https://arxiv.org/abs/2608.16884 · HTML: https://mlsanity.com/paper/2608.16884/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [A Refined Laser Method and Faster Matrix Multiplication](https://mlsanity.com/paper/2010.05846/) (2020)
- [New Bounds for Matrix Multiplication: from Alpha to Omega](https://mlsanity.com/paper/2307.07970/) (2023)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [More Asymmetry Yields Faster Matrix Multiplication](https://mlsanity.com/paper/2404.16349/) (2024)
- [Faster Matrix Multiplication via Asymmetric Hashing](https://mlsanity.com/paper/2210.10173/) (2022)
- [Supervised Quantile Normalization for Low-rank Matrix Approximation](https://mlsanity.com/paper/2002.03229/) (2020)

## Built on by (most cited first, 2 of 2)

- [A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations](https://mlsanity.com/paper/2609.03129/) (2026)
- [Change the Product, Keep the Parameters: Associative Algebra Layers for Transformers](https://mlsanity.com/paper/2609.32814/) (2026)

## Abstract (arXiv)

The current best bounds on the matrix multiplication exponent ω are obtained through a refinement of the laser method called combination loss analysis (Duan et al., 2022; Williams et al., 2024; Alman et al., 2025). In this note, we address the optimization problem at the core of this approach and propose several improvements. First, we reformulate the optimization problem allowing us to solve it in a larger setting than was previously possible. Second, we leverage recent advances in machine learning to design a new optimization algorithm for this problem. Finally, we refine the resulting optimization algorithm with AlphaEvolve. Our combined approach yields an upper bound of ω < 2.371177, improving the previous best bound of 2.371339.