🤖 AI Summary
This paper addresses the fragmentation and weak geometric intuition in existing kernel method theory by establishing a unified functional analytic framework grounded in Hilbert space geometry. Starting from the definition of positive-definite kernels, it rigorously unifies reproducing kernel Hilbert spaces (RKHS) and Hilbert–Schmidt operators, thereby reconstructing fundamental statistical concepts—including covariance, regression, and information-theoretic measures—within a coherent geometric setting. The work innovatively embeds kernel density estimation, distributional kernel embeddings, and maximum mean discrepancy (MMD) into a single RKHS paradigm, yielding a self-consistent theory bridging statistical estimation and probabilistic representation. This framework provides geometric interpretations for Gaussian processes and kernel Bayesian inference, and establishes a rigorous mathematical foundation for future theoretical advances in kernel-based learning.
📝 Abstract
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.