Learning functions, operators and dynamical systems with kernels

2025/09/22 by Lorenzo Rosasco, Rosasco, Lorenzo · 1 citation
Computer Science · Physics and Astronomy · #FOS: Computer and information sciences #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.2509.18071

openalex publication_date 2025/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Operator Learning: Algorithms and Analysis
• Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
• Koopman Operator, Geometry, and Learning of Dynamical Systems
• Randomized Numerical Linear Algebra: Foundations & Algorithms
• FALKON: An Optimal Large Scale Kernel Method
• Data-driven spectral decomposition and forecasting of ergodic dynamical\n systems
• Data-driven spectral decomposition and forecasting of ergodic dynamical systems
• Ridge Regression: Biased Estimation for Nonorthogonal Problems

Cited by

• An Operator Theoretical Approach to Mercer's Theorem

Related

• Operator-Valued Kernels, Machine Learning, and Dynamical Systems
• Notes on Kernel Methods in Machine Learning
• Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
• Operator-Based Machine Intelligence: A Hilbert Space Framework for Spectral Learning and Symbolic Reasoning
• Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nyström Method, and Use of Kernels in Machine Learning: Tutorial and Survey