Program Overview

Reproducing Kernel Hilbert Spaces with Applications

The course focuses on the reproducing kernel Hilbert spaces and applications, mostly on machine learning and data analysis. Although the data regression (fitting) and classification are at the heart of the content, we try to build a sound background of the mathematical structures related.

Course Objectives

At the end of this course, the student will learn:

• the fundamental function spaces used in applied mathematics
• kernels and kernel computations in related concepts
• the use of kernels in regression, classification, and related applications
• the use of kernels in differential and integral equations

Catalogue Content

Preparations: Linear Transformations and Matrices. Projections. Basics of Optimisation; Introduction: Finite-Dimensional RKHS. Normed. Metric. Inner Product and Hilbert Spaces; Concepts: Reproducing Kernel Hilbert Spaces. Kernels and kernel Computations; Applications in Machine Learning: Regression and Data Fitting. Classification. Maximum Mean Discrepancy (MMD). Hilbert-Schmidt Independence Criterion (HSIC); Gaussian Processes and Functional Data Analysis; Applications for Differential and Integral Equations: Ordinary, Partial and Integral Equations.

Learning Outcomes

Student, who passed the course satisfactorily will be able to:

• understand the fundamental function spaces and their basic properties
• cope with kernels, kernel computations, and constructing new kernels
• use kernels in mathematical and engineering applications, including machine learning

Suggested Textbooks

• Joe Suzuki, Kernel Methods for Machine Learning with Math and Python, Springer, 2022
• Vern I. Paulsen, Mrinal Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge University Press, 2016
• Saburou Saitoh, Yoshihiro Sawano, Theory of Reproducing Kernels and Applications, Springer, 2016
• Jonathon H. Manton, Pierre-Oliver Amblard, A Primer on Reproducing Kernel Hilbert Spaces, now Publishers Inc., 2015
• Alain Berlinet, Christine Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer Science+Business Media, 2004
• Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong, Mathematics for Machine Learning, Cambridge University Press, 2020