Know Your Boundaries: Constraining Gaussian Processes by Variational Harmonic Features
This work addresses a domain-specific challenge in regression and classification by incorporating physical information through boundary conditions, representing an incremental improvement over existing GP methods.
The paper tackles the problem of constraining Gaussian processes to arbitrarily-shaped domains with boundary conditions, achieving a low-rank representation that speeds up inference to O(nm^2) in prediction and O(m^3) in hyperparameter learning for regression.
Gaussian processes (GPs) provide a powerful framework for extrapolation, interpolation, and noise removal in regression and classification. This paper considers constraining GPs to arbitrarily-shaped domains with boundary conditions. We solve a Fourier-like generalised harmonic feature representation of the GP prior in the domain of interest, which both constrains the GP and attains a low-rank representation that is used for speeding up inference. The method scales as $\mathcal{O}(nm^2)$ in prediction and $\mathcal{O}(m^3)$ in hyperparameter learning for regression, where $n$ is the number of data points and $m$ the number of features. Furthermore, we make use of the variational approach to allow the method to deal with non-Gaussian likelihoods. The experiments cover both simulated and empirical data in which the boundary conditions allow for inclusion of additional physical information.