Kernel Methods for Machine Learning with Math and Python: 100 Exercises for Building Logic

✍ Scribed by Joe Suzuki

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 216
• Category: Library

✦ Synopsis

The most crucial ability for machine learning and data science is mathematical logic for grasping their essence rather than relying on knowledge or experience. This textbook addresses the fundamentals of kernel methods for machine learning by considering relevant math problems and building Python programs.

The book’s main features are as follows:

• The content is written in an easy-to-follow and self-contained style.
• The book includes 100 exercises, which have been carefully selected and refined. As their solutions are provided in the main text, readers can solve all of the exercises by reading the book.
• The mathematical premises of kernels are proven and the correct conclusions are provided, helping readers to understand the nature of kernels.
• Source programs and running examples are presented to help readers acquire a deeper understanding of the mathematics used.
• Once readers have a basic understanding of the functional analysistopics covered in Chapter 2, the applications are discussed in the subsequent chapters. Here, no prior knowledge of mathematics is assumed.
• This book considers both the kernel for reproducing kernel Hilbert space (RKHS) and the kernel for the Gaussian process; a clear distinction is made between the two.

✦ Table of Contents

Preface
How to Overcome Your Kernel Weakness
What Makes KMMP Unique?
Acknowledgments
Contents
1 Positive Definite Kernels
1.1 Positive Definiteness of a Matrix
1.2 Kernels
1.3 Positive Definite Kernels
1.4 Probability
1.5 Bochner's Theorem
1.6 Kernels for Strings, Trees, and Graphs
Appendix
Exercises 1 sim 15
2 Hilbert Spaces
2.1 Metric Spaces and Their Completeness
2.2 Linear Spaces and Inner Product Spaces
2.3 Hilbert Spaces
2.4 Projection Theorem
2.5 Linear Operators
2.6 Compact Operators
Appendix: Proofs of Propositions
Exercises 16 sim 30
3 Reproducing Kernel Hilbert Space
3.1 RKHSs
3.2 Sobolev Space
3.3 Mercer's Theorem
Appendix
Exercises 31 sim 45
4 Kernel Computations
4.1 Kernel Ridge Regression
4.2 Kernel Principle Component Analysis
4.3 Kernel SVM
4.4 Spline Curves
4.5 Random Fourier Features
4.6 Nyström Approximation
4.7 Incomplete Cholesky Decomposition
Appendix
Exercises 46 sim 64
5 The MMD and HSIC
5.1 Random Variables in RKHSs
5.2 The MMD and Two-Sample Problem
5.3 The HSIC and Independence Test
5.4 Characteristic and Universal Kernels
5.5 Introduction to Empirical Processes
Appendix
Exercises 65 sim83
6 Gaussian Processes and Functional Data Analyses
6.1 Regression
6.2 Classification
6.3 Gaussian Processes with Inducing Variables
6.4 Karhunen-Lóeve Expansion
6.5 Functional Data Analysis
Appendix
Exercises 83sim100
Appendix Bibliography