Researchers from Google DeepMind, Carnegie Mellon, MIT, and Columbia University have established a new upper bound for the matrix multiplication exponent $\omega$, reporting it as less than 2.371177. This figure improves upon the previous best known bound of 2.371339. The work, detailed in a paper published on arXiv, leverages a reformulated optimization problem, a new machine learning-based optimization algorithm, and refinement with AlphaEvolve, an AI-powered coding agent.
The matrix multiplication exponent $\omega$ describes the asymptotic complexity of multiplying two $n \times n$ matrices. A lower value for $\omega$ indicates greater computational efficiency for large-scale matrix operations. The theoretical minimum for $\omega$ is 2, though this has not yet been achieved. Each reduction in the upper bound of $\omega$ often requires novel mathematical techniques and can have implications for various fields, including linear algebra, graph algorithms, and other computations that rely on matrix products.
The current best bounds on $\omega$ have been derived through a method known as combination loss analysis, a refinement of the "laser method." Previous work by Duan et al. (2022), Williams et al. (2024), and Alman et al. (2025) contributed to the prior record. The new research addresses the core optimization problem within this approach by introducing several improvements.
First, the researchers reformulated the optimization problem, which enabled its solution in a larger setting than was previously possible. This expanded search space allowed for the exploration of a broader range of potential solutions. Second, they designed a new optimization algorithm by incorporating recent advancements in machine learning. This algorithm provided a more effective means of navigating the complex landscape of the reformulated problem.
The final step involved refining this new optimization algorithm using AlphaEvolve. AlphaEvolve, developed by Google DeepMind and unveiled in May 2025, is an evolutionary coding agent that utilizes large language models (LLMs) and evolutionary computation to discover and refine algorithms. It is designed as a general-purpose system capable of operating across various scientific and engineering tasks by modifying code and optimizing for multiple objectives.
Google DeepMind has previously highlighted AlphaEvolve's capabilities in designing faster matrix multiplication algorithms. In 2025, AlphaEvolve found an algorithm to multiply 4x4 complex-valued matrices using 48 scalar multiplications, improving upon Strassen's 1969 algorithm which used 49. While this specific improvement was for a fixed-size matrix, such reductions in base cases can asymptotically propagate to the complexity exponent $\omega$ for large matrices.
AlphaEvolve's design allows it to autonomously discover and refine algorithms by using LLMs to generate variants of existing algorithms and then selecting the most effective ones based on an evaluation function. This system has been applied to over 50 open mathematical problems, rediscovering state-of-the-art solutions in 75% of cases and discovering improved solutions in 20%. It has also been used to optimize Google's computing infrastructure, including data center scheduling heuristics and the design of Tensor Processing Units (TPUs).
The research paper lists ten authors, including Josh Alman and Virginia Vassilevska Williams, who were part of the team that established the previous $\omega < 2.371339$ record. Their involvement suggests this new result represents a direct advancement within the leading edge of the field. The improvement of approximately 0.000162, while numerically small, signifies a continued push towards the theoretical lower bound of 2 for the matrix multiplication exponent.
The long-term implications of reducing the matrix multiplication exponent extend to any computational problem that can be reduced to matrix multiplication. As computational demands continue to grow, particularly with the rise of large AI models, even small theoretical improvements in fundamental operations like matrix multiplication can lead to substantial efficiency gains in practice. AlphaEvolve became generally available on Google Cloud in July 2026, making its optimization capabilities accessible for broader research and business applications.