An $O(n^2\log^4 n \log \log n)$ Time Matrix Multiplication Algorithm

TL;DR

This paper presents a novel matrix multiplication algorithm that reduces the computational complexity to approximately $O(n^2 ext{polylog}(n))$, significantly improving over traditional methods by leveraging advanced preprocessing techniques.

Contribution

The paper introduces a new approach that achieves $O(n^2 ext{polylog}(n))$ time for matrix multiplication through efficient preprocessing of vectors.

Findings

Achieves $O(n^2 ext{polylog}(n))$ matrix multiplication time.

Preprocessing each vector takes $O(n ext{polylog}^4 n)$ time.

Enables faster matrix computations in theoretical and practical applications.

Abstract

We show, for the input vectors $(a_0, a_1, ..., a_{n-1})$ and $(b_0, b_1, ..., b_{n-1})$, where $a_i$'s and $b_j$'s are real numbers, after $O(n\log^4 n)$ time preprocessing for each of them, the vector multiplication $(a_0, a_1, ..., a_{n-1})(b_0, b_1, ..., b_{n-1})^T $ can be computed in $O(\log^4 n \log \log n)$ time. This enables the matrix multiplication for two $n\times n$ matrices to be computed in $O(n^2 \log^4 n\log \log n)$ time.