[PyMCon Web Series 03a] Introduction to Hilbert Space GPs (HSGPs) in PyMC (Mar 15, 2023) (Bill Engels)

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I hope it helps!

  1. Is HSGP also working with WrapedInput kernels? I’m frequently using WrapedInput to model GPs on non-euclidean domains, e.g. data on a 2-D sphere.

Yes and no. You can absolutely transform your X inputs, but you’ll have to do it outside of the WarpedInput covariance kernel unfortunately. HSGPs can only be applied to stationary kernels with a power spectral density. So for now, ExpQuad, Matern52 and Matern32.

  1. I’m not an expert on Hilbert space methods, but if I understood your approach correctly, HSGP is basically approximating the GP via a Fourier series with random coefficients. Then, it should be quite natural to extend this method to use any kind of orthonormal basis functions for approximation, e.g. orthogonal polynomials, wavelets, etc. - shouldn’t it? Intuitively, I would have maybe chosen Hermite polynomials to approximate GPs on an euclidean domain, to avoid the artificial periodicity that seems to be currently introduced if you predict values outside [−L,L].

It’s certainly not my method! I merely ported it into PyMC. It was first proposed in this paper. I don’t think you can trivially extend the method to use any type of orthonormal basis functions because it depends on the expression of a kernel as a power spectral density.