Fast Gaussian Process Predictions on Large Geospatial Fields with Prediction-Point Dependent Basis Functions

In order to perform GP predictions fast in large geospatial fields with small-scale variations, a computational complexity that is independent of the number of measurements $N$ and the size of the field is crucial. In this setting, GP approximations using $m$ basis functions requires $\mathcal{O}(Nm^2+m^3)$ computations. Using finite-support basis functions reduces the required number of computations to perform a single prediction to $\mathcal{O}(m^3)$, after a one-time training cost of $O(N)$. The prediction cost increases with increasing field size, as the number of required basis functions $m$ grows with the size of the field relative to the size of the spatial variations. To prevent the prediction speed from depending on field size, we propose leveraging the property that a subset of the trained system is a trained subset of the system to use only a local subset of $m'\ll m$ finite-support basis functions centered around each prediction point to perform predictions. Our proposed approximation requires $\mathcal{O}(m'^3)$ operations to perform each prediction after a one-time training cost of $\mathcal{O}(N)$. We show on real-life spatial data that our approach matches the prediction error of state-of-the-art methods and that it performs faster predictions, also compared to state-of-the-art approximations that lower the prediction cost of $\mathcal{O}(m^3)$ to $\mathcal{O}(m\log(m))$ using a conjugate gradient solver. Finally, we demonstrate that our approach can perform fast predictions on a global bathymetry dataset using millions of basis functions and tens of millions of measurements on a laptop computer.