[论文解读] Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

Motonobu Kanagawa, Philipp Hennig+2 more|arXiv (Cornell University)|Jul 6, 2018

Gaussian Processes and Bayesian Inference参考文献 70被引用 193

一句话总结

本论文综述高斯过程（GP）贝叶斯方法与再现核希尔伯特空间（RKHS）核方法之间的深层联系，澄清等价性与差异，以促进跨领域结果的转移。

ABSTRACT

This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.

研究动机与目标

• Bridge conceptual gaps between Bayesian GP inference and frequentist RKHS kernel methods.
• Clarify when GP posterior quantities correspond to kernel method quantities and interpret posterior variance in a frequentist sense.
• Discuss how hypothesis spaces differ between GP priors and RKHS, and what this implies for modeling and analysis.
• Present connections in convergence, posterior contraction, and integral transforms to enable cross-disciplinary transfer of results.
• Provide pedagogical overview for researchers new to either field.

提出的方法

• Review and juxtapose algorithmic quantities from GP and RKHS frameworks to highlight similarities.
• Explain equivalences such as GP posterior mean matching kernel ridge regression estimators.
• Show posterior variance as a worst-case error in RKHS and discuss noise/additive regularization equivalences.
• Use spectral representations (Mercer, Karhunen–Loève) to compare GP draws and RKHS functions.
• Discuss convergence rates and contraction results by relating GP-based analyses to RKHS-based analyses.

实验结果

研究问题

• RQ1 How are Gaussian process regression and kernel ridge regression equivalent in terms of estimators and regularization?
• RQ2 In what sense can GP posterior variance be interpreted as a worst-case error in RKHS-based regression?
• RQ3 What are the precise relationships between GP draws and RKHS hypothesis spaces, particularly regarding sample paths and smoothness?
• RQ4 How do convergence and posterior contraction rates for GP regression relate to those for kernel ridge regression?
• RQ5 How do integral transforms, such as kernel mean embeddings and HSIC, connect GP and RKHS perspectives?

主要发现

• Posterior mean in GP regression coincides with the kernel ridge regression estimator when using the same kernel.
• Posterior variance in GP regression corresponds to a worst-case error in RKHS, linking average-case and worst-case analyses.
• Regularization in kernel ridge regression and additive Gaussian noise in GP regression play analogous roles in smoothing and bias-variance trade-offs.
• GP sample paths almost surely lie outside the RKHS, but reside in a larger function space related to the RKHS, enabling cross-framework insights.
• Convergence rates for GP-regression can be recovered from kernel ridge regression results by considering approximate embeddings in slightly larger spaces, with noise variance behavior connected to regularization schedules.