Towards Faster Feasible Matrix Multiplication by Trilinear Aggregation

TL;DR

This paper introduces a new fast matrix multiplication algorithm using trilinear aggregation, improving asymptotic complexity and practical applicability for small to medium-sized matrices.

Contribution

The authors develop an improved algorithm with a lower exponent and better base case complexity, advancing the practical efficiency of fast matrix multiplication methods.

Findings

Achieved an $O(n^{2.773203})$ algorithm applicable to feasible matrix sizes.

Optimized the algorithm for small base cases starting at $n_0=28$.

Reduced additive complexity through sparse decomposition and coefficient minimization.

Abstract

Matrix multiplication is a fundamental kernel in high performance computing. Many algorithms for fast matrix multiplication can only be applied to enormous matrices ($n>10^{100}$) and thus cannot be used in practice. Of all algorithms applicable to feasible input, Pan's $O(n^{2.773372})$ algorithm (1982) is asymptotically the fastest. We obtain an $O(n^{2.773203})$ algorithm applicable to the same input sizes as Pan's algorithm. This algorithm is the fastest matrix multiplication algorithm with base case smaller than $1000$. Further, our method obtains the best asymptotic complexity for many small base cases, starting at $n_0=28$. We also obtain better exponents for larger base cases. To construct our algorithm, we use the trilinear aggregation method. We find parts of the algorithms that are equivalent to matrix multiplication with smaller base case, and use the de Groote equivalence to replace these parts in a way that allows further optimization of our algorithms. Finally, we improve the additive complexity of our algorithms by finding a sparse decomposition and reducing the leading coefficient. These mark a fundamental step towards outperforming existing fast matrix multiplication algorithms in practice.