揭示高斯过程与再生核希尔伯特空间的深层等价关系，统一两类方法的核心原理。
Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences
- 基于高斯希尔伯特空间与RKHS的等价性，建立统一视角。
- 在回归、插值、积分等任务中发现两类方法的数学等价性。
- 适合研究机器学习理论或核方法的学者参考。
本专著研究了正定核在两类方法中的应用：基于高斯过程的概率方法与基于再生核希尔伯特空间（RKHS）的非概率方法。这两类方法在机器学习、统计学和数值分析中被广泛研究与应用。本文探讨了回归、插值、数值积分、分布差异度量及统计依赖性等基本问题中的联系与等价性，同时研究了高斯过程的样本路径性质。通过高斯希尔伯特空间与RKHS之间的等价性，建立了一个统一的理论框架，为连接由两个研究领域分别发展的多种基于高斯过程与再生核的方法奠定了基础。
原文摘要 · Abstract (English)
This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. We study connections and equivalences for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.
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