Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

David Maslan,Daniel N Rockmore,Sarah Wolff

doi:10.46298/dmtcs.6372

David Maslan, Daniel N Rockmore + Show 1 more

Open Access

https://doi.org/10.46298/dmtcs.6372

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Abstract

We present a general diagrammatic approach to the construction of efficient algorithms for computingthe Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to theconstruction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work inthe setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to pathcounting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Ourmethods give improved upper bounds for computing the Fourier transform for the general linear groups over finitefields, the classical Weyl groups, and homogeneous spaces of finite groups.

Highlights

The Fast Fourier Transform (FFT) remains among the most important family of algorithms in information processing [Roc00]. It efficiently computes the discrete Fourier transform (DFT) which is equivalent to the matrix-vector multiplication e2πijk/n f for i = −1, j, k = 0, . . . n − 1, and f a complex-valued vector of length n [Roc00]
Sarah Wolff change of basis in C[CN ], the complex group algebra of the cyclic group of order N, from a natural basis of group element indicator functions to a basis of irreducible matrix elements. This perspective suggests a generalization of the DFT to finite nonabelian groups G as the computation of a change of basis in C[G] from a basis of indicator functions to a basis of irreducible matrix elements and raises attendant questions of computational complexity addressed
The fundamental idea of the SOV approach is a recasting of the Cooley-Tukey algorithm in terms of graded quivers, which is an elaboration of path algebras derived from Bratteli diagrams

Summary

Introduction

The Fast Fourier Transform (FFT) remains among the most important family of algorithms in information processing [Roc00] It efficiently computes the discrete Fourier transform (DFT) which is equivalent to the matrix-vector multiplication e2πijk/n f (1). It is a divide-and-conquer algorithm whose key step is to rewrite the DFT on a cyclic group CN as a linear combination of DFTs on Cn < CN (for n | N ) Iterating this step for a chain of subgroups of CN yields algorithms more efficient than a direct matrix-vector multiplication. The “repeated units” of our divideand-conquer approach are subgraphs of a Bratteli diagram and efficiencies are gained by recognizing their multiple appearances in the corresponding calculation. This is the guts of the “separation of variables”.

Fourier transforms and the group algebra

Bratteli diagrams and quivers

The Separation of Variables Approach

Morphisms into Bratteli Diagrams