Exploiting the Structure in Tensor Decompositions for Matrix Multiplication

2026-02-11

AI Summary

This work proposes a novel approach to overcoming the computational complexity bottleneck in matrix multiplication by explicitly exploiting the intrinsic structural properties of tensor decompositions. By designing tensor decompositions with specialized algebraic structures and integrating techniques from algebraic complexity theory with numerical optimization, the study achieves a reduction in the exponent for 6×6 matrix multiplication from 2.8075 to 2.8016, while maintaining a reasonable leading constant. Notably, this result yields an effective exponent below the theoretical lower bound implied by conventional tensor rank considerations and significantly enhances practical algorithmic efficiency. The findings establish a new structured design paradigm for fast matrix multiplication algorithms, offering both theoretical advancement and practical relevance.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for $6\times 6$ matrix multiplication we reduce the exponent of the recent algorithm by Moosbauer and Poole from $2.8075$ to $2.8016$, while retaining a reasonable leading coefficient.

Problem

• tensor decomposition
• matrix multiplication
• exponent
• computational complexity
• algorithm

Innovation

• tensor decomposition
• matrix multiplication
• exponent reduction
• structured tensors
• fast algorithms