Kernel-Based Methods: Milestones and New Trends(Series Number 46 Cambridge Monographs on Applied and Computational Mathematics)

Kernel methods, with origins in the pioneering work of Mercer (1909), Bochner (1933), and Aronszajn (1950), have become central tools in modern mathematics and machine learning. This book explores their deep connections with approximation theory, highlighting both classical results and cutting-edge developments. Through clear explanations and illustrative examples, it guides readers from foundational concepts to contemporary applications, including computational methods and real-world problem solving. By bridging theory and practice, the text not only provides a rigorous understanding of kernels but also inspires further exploration and research. Suitable for students, researchers, and practitioners, it invites readers to engage with ongoing advances in this dynamic field and to contribute to its future growth.

Table of Contents:
List of illustrations; List of tables; 1. Introduction; 2. Positive definite kernels and functions; 3. RBF interpolation and polynomial precision; 4. Error estimates and examples of kernels; 5. On uncertainty principles and shape parameter selection; 6. Rational kernels; 7. Partition of Unity; 8. Efficient reduced-basis approximation; 9. Approximating via variably scaled discontinuous kernels; 10. Application of VSDKs to magnetic particle imaging; 11. Applications to astronomical imaging; 12. Kernels in machine learning; 13. Support vector machines and other methods; 14. Local-to-Global SVMs; 15. Learning data-driven optimized kernels; 16. Choosing and learning the scaling function for the VSKs; 17. Kernels for persistent homology; 18. Software; 19. Conclusions; Bibliography; References; Index.

About the Author :
Stefano De Marchi is Full Professor of Numerical Analysis at the University of Padova. He has authored more than 150 publications and is the managing editor of the open-access journal Dolomites Research Notes on Approximation. He is the founder of the Italian Network on Approximation and one of the discoverers of the so-called Padua points. Francesco Marchetti is Assistant Professor of Numerical Analysis at the University of Padova. His research lies at the intersection of approximation theory and machine learning, focusing on kernel methods and polynomial and RBF interpolation, with applications to medical imaging and space weather forecasting, within national and international research projects. Emma Perracchione is Associate Professor at Politecnico di Torino. Her research focuses on approximation theory and its applications to solar physics, such as astronomical imaging. She is currently Principal Investigator of the GOSSIP project (Greedy Optimal Sampling for Solar Inverse Problems), funded by the Italian Ministry of Universities and Research.

Review :
'This is an extremely helpful book for researchers who work with kernels in the context of modern applications – including machine learning, image processing, and data science. It provides important new theoretical results as well as numerous practical examples, and supplies the necessary software to open up new kernel-based fields of research.' Robert Schaback, University of Göttingen 'This volume complements the existing textbooks on kernel methods with exciting new topics such as rational kernel approximation, approximation of discontinuous data, and learning with kernels.' Elisabeth Larsson, Uppsala University 'Kernel and radial basis function-based approximations are at the very centre of modern multivariate approximation theory and applications. This book reviews state-of-the-art approaches including excellent summaries of the basic theoretical results on kernel-based methods and their applications; the practical aspects of these tools are particularly emphasized and a great number of very useful tools, including software, is included. This is a book that should be in every application-interested scientist who works with multivariate approximations.' Martin Buhmann, University of Giessen 'This monograph complements existing books on kernels and their applications by focusing on promising recent developments – such as rational kernels, data-driven kernels, variably scaled kernels, persistence kernels – the details of which were previously accessible only in original research publications.' Oleg Davydov, University of Giessen 'This book seamlessly bridges rigorous mathematical theory with practical, hands-on applications. Readers will explore cutting-edge solutions for scattered data problems – with a special focus on variably scaled kernels (VSKs) – alongside imaging challenges and vital machine learning tools like support vector machines (SVMs). To ensure immediate implementation, the text introduces key software packages with direct links to open-source code repositories on GitHub. Designed for students, applied scientists, and practitioners alike, this text bypasses advanced functional analysis to deliver a direct, comprehensive understanding of kernel-based algorithms.' Greg Fasshauer, Colorado School of Mines