Bispectrum on finite groups

The algebraic theory of finite groups appears in signal processing problems involving the statistical analysis of ranked data and the construction of invariants for pattern recognition. Standard signal processing techniques involving spectral analysis are, in theory, possible for data defined on finite groups by using the Fourier transform provided by group representations. However, one such technique, the bispectrum, which is useful for analysing non-Gaussian data as well as for constructing geometric invariants, has not been explored in detail for finite groups. This paper shows how to construct the bispectrum on an arbitrary finite group or homogeneous space and explores its properties. Examples are given using the symmetric group as well as wreath-product groups.