The skew spectrum of functions on finite groups and their homogeneous spaces

TL;DR

This paper introduces the skew spectrum, a computationally efficient set of invariants for functions on finite groups and homogeneous spaces, offering a lossless and invariant representation that simplifies bispectrum calculations.

Contribution

The paper proposes the skew spectrum as an easier-to-compute alternative to the bispectrum for functions on groups and homogeneous spaces, utilizing Clausen-type FFT techniques.

Findings

Skew spectrum is unitarily equivalent to the bispectrum.

Efficient computation of the skew spectrum using fast Fourier transforms.

Provides invariant representations for functions on groups and spaces.

Abstract

Whenever we have a group acting on a class of functions by translation, the bispectrum offers a principled and lossless way of representing such functions invariant to the action. Unfortunately, computing the bispectrum is often costly and complicated. In this paper we propose a unitarily equivalent, but easier to compute set of invariants, which we call the skew spectrum. For functions on homogeneous spaces the skew spectrum can be efficiently computed using some ideas from Clausen-type fast Fourier transforms.