GPs in the Fourier domain

The day started with Dun Wang, Steven Mohammed (Columbia), David Schiminovich and I discussing the short-term plans for our work with GALEX. My top priority is to get the flat-field right, because if we can do that, I think we will be able to do everything else (pointing model, focal-plane distortion model, etc.).

Over lunch, Greengard and Jeremy Magland (SCDA) “reminded me” how the FFT works in the case of irregularly sampled data. This in the context of using Gaussian-process kernels built not in real space but in Fourier space. And then Greengard and Magland more-or-less simultaneously suggested that maybe we can turn all our Gaussian process problems into convolution problems! The basic idea is that the matrix product of a kernel matrix and a vector looks very close to a convolution, and the product with the inverse matrix looks like a deconvolution. And we know how to do this fast in Fourier space. This could be huge for asteroseismology. The log-determinant may also be simple when we think about it all in Fourier space. We will reconvene this conversation late next week.


  1. We just met with statistician Chris Paciorek at UC Berkeley on 2015-10-15 about sampling from very large GPs for modeling lensing shear and convergence in our probabilistic cosmic shear framework. I learned about Gaussian Markov Random Fields and associated sparse precision matrices for Matern covariances. This sounds similar to what you're describing here?

    1. Michael: I've been shouting in the desert about this (the GMRF to GP connection via SPDEs) for a while now but no-one in astro is listening (see e.g. http://arxiv.org/abs/1406.6371 and a previous use in cosmology by some Dubliners here: http://arxiv.org/abs/1011.4018 )! Non-stationarity = solved. GPs on manifolds = solved. Fast inference with INLA = check. etc. etc. Would be very interested to talk with you if you have some applications in mind ...