Adaptive Resolution Framework for Finite-Rank Gaussian Processes Achieves Optimal Contraction Rates
Researchers have developed a theoretical framework showing that finite-rank Gaussian process approximations can achieve the same posterior contraction rates as their full parent GP priors when a suitable prior is placed on the resolution parameter. The work covers two widely used approximation families—finite-element methods based on the SPDE representation of Matérn GPs and regular-grid interpolation schemes—and also introduces a practical posterior sampler that jointly updates resolution and bandwidth. This matters because it provides rigorous statistical justification for scalable GP methods, including minimax-optimal convergence rates, which had previously lacked such guarantees.
Gaussian process regression is a powerful but computationally expensive tool, and finite-rank approximations are commonly used to make it tractable at scale. However, these approximations can behave differently from the full GP prior in their posterior, raising questions about their statistical validity. This paper by Jaehoan Kim and collaborators addresses that gap by studying finite-rank GP priors built from locally supported basis expansions with dependent Gaussian coefficients. The authors prove that placing a suitable prior on the resolution parameter N causes these approximations to inherit the posterior contraction rates of the parent GP. Specifically, the interpolation construction under a squared-exponential parent GP achieves the minimax-optimal rate up to logarithmic factors under a hierarchical prior on both bandwidth and N, while the SPDE-based construction achieves the same rate under a bandwidth scaling tied to sample size and function smoothness. The paper also contributes a practical posterior sampler for the hierarchical interpolation model and supports the theoretical results with numerical experiments across five figures in a 48-page manuscript.
What's missing
It is unclear whether the theoretical guarantees extend to non-Gaussian likelihoods or classification settings, and the computational cost of the proposed joint sampler relative to standard sparse GP methods is not assessed in the abstract.
What different sources said
- arXiv stat.MLCenter
Adaptive Resolution for Finite-Rank Gaussian Processes
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