Adaptive Resolution for Finite-Rank Gaussian Processes

Finite-rank approximations are widely used to scale Gaussian process (GP) regression, but their posterior behavior can differ from that of the corresponding parent GP prior. We study a class of finite-rank GP priors built from locally supported basis expansions with dependent Gaussian coefficients. Our framework covers finite-element approximations based on the stochastic partial differential equation (SPDE) representation of Mat'ern GPs and regular-grid GP interpolation schemes. We show that, with a suitable prior on the resolution parameter N, these finite-rank expansions inherit the same posterior contraction rate as the corresponding parent GP prior under the same bandwidth specification used for that parent prior. Consequently, the interpolation construction under a squared-exponential parent GP attains the minimax-optimal rate up to logarithmic factors under a hierarchical prior on the bandwidth parameter and on N, while the SPDE construction attains the same rate under a bandwidth scaling depending on the sample size and the smoothness of the true function, together with a prior on N. We also develop a posterior sampler for the hierarchical interpolation model that jointly updates the resolution and bandwidth parameters, and we provide numerical studies that support the theory.