Combining Pseudo-Point and State Space Approximations for Sum-Separable Gaussian Processes

Gaussian processes (GPs) are important probabilistic tools for inference and learning in spatio-temporal modelling problems such as those in climate science and epidemiology. However, existing GP approximations do not simultaneously support large numbers of off-the-grid spatial data-points and long time-series which is a hallmark of many applications. Pseudo-point approximations, one of the gold-standard methods for scaling GPs to large data sets, are well suited for handling off-the-grid spatial data. However, they cannot handle long temporal observation horizons effectively reverting to cubic computational scaling in the time dimension. State space GP approximations are well suited to handling temporal data, if the temporal GP prior admits a Markov form, leading to linear complexity in the number of temporal observations, but have a cubic spatial cost and cannot handle off-the-grid spatial data. In this work we show that there is a simple and elegant way to combine pseudo-point methods with the state space GP approximation framework to get the best of both worlds. The approach hinges on a surprising conditional independence property which applies to space--time separable GPs. We demonstrate empirically that the combined approach is more scalable and applicable to a greater range of spatio-temporal problems than either method on its own.