Adaptive Resolution for Finite-Rank Gaussian Processes

2025/05/29 by Jaehoan Kim, Kim, Jaehoan, Anirban Bhattacharya +3
Engineering · #Advanced Measurement and Metrology Techniques #Advanced Numerical Analysis Techniques #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Methodology (stat.ME) #Reservoir Engineering and Simulation Methods #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2505.24066

openalex publication_date 2025/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Matérn 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.

Citations

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