Fast generalized DFTs for all finite groups

TL;DR

This paper introduces an efficient arithmetic algorithm for computing generalized Discrete Fourier Transforms over any finite group, significantly improving computational complexity by leveraging matrix multiplication exponents.

Contribution

It provides the first general algorithm for all finite groups with complexity tied to the matrix multiplication exponent, advancing computational harmonic analysis.

Findings

Achieves $O(|G|^{rac{ ext{omega}}{2} + ext{epsilon}})$ complexity for all finite groups.

Extends the applicability of fast Fourier transform techniques to arbitrary finite groups.

Bridges group theory and matrix multiplication complexity for efficient Fourier computations.

Abstract

For any finite group $G$, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to $G$, using $O(|G|^{\omega/2 + \epsilon})$ operations, for any $\epsilon > 0$. Here, $\omega$ is the exponent of matrix multiplication.