从几何视角解析核方法，打通机器学习理论基础
Notes on Kernel Methods in Machine Learning
- 基于希尔伯特空间构建核方法的数学框架
- 提出概率分布的核嵌入与最大均值差异度量
- 适合想深入理解核方法原理的研究者
这些笔记提供了核方法及其在机器学习中几何基础的自包含介绍。从希尔伯特空间构造出发，系统阐述正定核、再生核希尔伯特空间（RKHS）及希尔伯特-施密特算子的理论，强调其在统计估计和概率测度表示中的作用。经典概念如协方差、回归与信息度量通过希尔伯特空间几何重新审视。引入核密度估计、分布的核嵌入及最大均值差异（MMD）。内容旨在为高斯过程、核贝叶斯推断及现代机器学习的功能分析方法奠定基础。
原文摘要 · Abstract (English)
These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing kernel Hilbert spaces (RKHS), and Hilbert-Schmidt operators, emphasizing their role in statistical estimation and representation of probability measures. Classical concepts such as covariance, regression, and information measures are revisited through the lens of Hilbert space geometry. We also introduce kernel density estimation, kernel embeddings of distributions, and the Maximum Mean Discrepancy (MMD). The exposition is designed to serve as a foundation for more advanced topics, including Gaussian processes, kernel Bayesian inference, and functional analytic approaches to modern machine learning.
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