Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the Compact Case

by   Iskander Azangulov, et al.

Gaussian processes are arguably the most important model class in spatial statistics. They encode prior information about the modeled function and can be used for exact or approximate Bayesian inference. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners.


page 3

page 10

page 13

page 15

page 21

page 23


Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces

Gaussian processes are arguably the most important class of spatiotempor...

Gaussian Processes and Statistical Decision-making in Non-Euclidean Spaces

Bayesian learning using Gaussian processes provides a foundational frame...

Matern Gaussian processes on Riemannian manifolds

Gaussian processes are an effective model class for learning unknown fun...

Isotropic Gaussian Processes on Finite Spaces of Graphs

We propose a principled way to define Gaussian process priors on various...

Random walk kernels and learning curves for Gaussian process regression on random graphs

We consider learning on graphs, guided by kernels that encode similarity...

Replica theory for learning curves for Gaussian processes on random graphs

Statistical physics approaches can be used to derive accurate prediction...

Scale invariant process regression

Gaussian processes are the leading method for non-parametric regression ...

Please sign up or login with your details

Forgot password? Click here to reset