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Gaussian processes (GPs) are important probabilistic tools for inference andlearning in spatio-temporal modelling problems such as those in climate scienceand epidemiology. However, existing GP approximations do not simultaneouslysupport large numbers of off-the-grid spatial data-points and long time-serieswhich is a hallmark of many applications.  Pseudo-point approximations, one of the gold-standard methods for scaling GPsto large data sets, are well suited for handling off-the-grid spatial data.However, they cannot handle long temporal observation horizons effectivelyreverting to cubic computational scaling in the time dimension. State space GPapproximations are well suited to handling temporal data, if the temporal GPprior admits a Markov form, leading to linear complexity in the number oftemporal observations, but have a cubic spatial cost and cannot handleoff-the-grid spatial data.  In this work we show that there is a simple and elegant way to combinepseudo-point methods with the state space GP approximation framework to get thebest of both worlds. The approach hinges on a surprising conditionalindependence property which applies to space--time separable GPs. Wedemonstrate empirically that the combined approach is more scalable andapplicable to a greater range of spatio-temporal problems than either method onits own.

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