Kernel-Based Methods

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.

'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

• 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.