Adaptive Resolution Framework for Finite-Rank Gaussian Processes Achieves Optimal Contraction Rates
Researchers developed a theoretical framework showing that finite-rank Gaussian process approximations can achieve the same posterior contraction rates as their parent GP priors when equipped with suitable priors on the resolution parameter. The work covers finite-element approximations based on SPDE representations of Matérn GPs and regular-grid interpolation schemes. This result is significant because it provides theoretical justification for using computationally efficient finite-rank approximations without sacrificing statistical optimality.
The paper addresses a fundamental challenge in scaling Gaussian process regression: finite-rank approximations are computationally efficient but their posterior behavior can diverge from the parent GP prior. The authors study a class of finite-rank GP priors built from locally supported basis expansions with dependent Gaussian coefficients, encompassing both SPDE-based finite-element methods for Matérn GPs and regular-grid interpolation schemes. Under appropriate hierarchical priors on resolution and bandwidth parameters, they prove that these approximations inherit minimax-optimal posterior contraction rates (up to logarithmic factors) matching their parent priors. The theoretical results are complemented by a posterior sampler that jointly updates resolution and bandwidth parameters, with numerical studies validating the theoretical predictions.
What's missing
The paper does not discuss computational complexity comparisons between the proposed adaptive resolution approach and existing finite-rank approximation methods, nor does it provide guidance on practical hyperparameter selection beyond the theoretical framework. Additionally, the scope appears limited to regression settings; applicability to classification or other GP tasks is not addressed.
What different sources said
- arXiv stat.MLCenter
Adaptive Resolution for Finite-Rank Gaussian Processes
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