I'm familiar with Risi's work on using Fourier transform on groups for object tracking. Basic idea is that the usual Fourier transform is a construction on Z_n, which generalizes to antsy finite groups. Important examples are binary vectors (Z_2n) and permutations. The twin hypotheses are that learning on Fourier space is easier, and that learning on these other specialized groups for which we understand the transform is a useful pursuit.

For the permutation group, you get matrices instead of numbers for the Fourier coefficients. The matrices themselves have nice interpretations (eg, interactions of unordered pairs or ordered triples of elements in the permutation), but the actual entries of the matrices are all but impossible to motivate, imho. (Much like how it's easier to understand the magnitude of a regular Fourier coefficient than the phase.)

A related article that some might find interesting:

Why does Deep Learning work? - A perspective from Group Theory -- https://arxiv.org/abs/1412.6621

groupMembership :: Perm -> Bool

semigroupMembership :: Endo -> Bool

The latter is much more interesting.

Would love to hear people's thoughts on this. I don't have the mathematical background to understand this work.

Very good and interesting work!