Integrated Variational Fourier Features for Fast Spatial Modelling with Gaussian Processes

Sparse variational approximations are popular methods for scaling up inference and learning in Gaussian processes to larger datasets. For N training points, exact inference has O(N3) cost; with M ≪ N features, state of the art sparse variational methods have O(NM2) cost. Recently, methods have been proposed using more sophisticated features; these promise O(M3) cost, with good performance in low dimensional tasks such as spatial modelling, but they only work with a very limited class of kernels, excluding some of the most commonly used. In this work, we propose integrated Fourier features, which extends these performance benefits to a very broad class of stationary covariance functions. We motivate the method and choice of parameters from a convergence analysis and empirical exploration, and show practical speedup in synthetic and real world spatial regression tasks.