Learning functions, operators and dynamical systems with kernels
It serves as educational material for a course, providing a foundational review without introducing new methods or results.
The paper presents an expository overview of statistical machine learning using reproducing kernel Hilbert spaces, extending the framework from scalar-valued learning to operator learning and applying it to dynamical systems via Koopman operator theory.
This expository article presents the approach to statistical machine learning based on reproducing kernel Hilbert spaces. The basic framework is introduced for scalar-valued learning and then extended to operator learning. Finally, learning dynamical systems is formulated as a suitable operator learning problem, leveraging Koopman operator theory. The manuscript collects the supporting material for the corresponding course taught at the CIME school "Machine Learning: From Data to Mathematical Understanding" in Cetraro.