Nonparametric Sparse Online Learning of the Koopman Operator

TL;DR

This paper introduces an online sparse learning algorithm for the Koopman operator in RKHS, addressing the challenge of model mis-specification and providing theoretical guarantees and numerical validation.

Contribution

It develops a novel operator stochastic approximation method for Koopman learning in RKHS, with convergence guarantees under mis-specified dynamics.

Findings

Algorithm effectively learns Koopman operators from data.

Theoretical guarantees for convergence and complexity.

Numerical experiments demonstrate practical performance.

Abstract

The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. Data-driven techniques to learn the Koopman operator typically assume that the chosen function space is closed under system dynamics. In this paper, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and explore the mis-specified scenario where the dynamics may escape the chosen function space. We relate the Koopman operator to the conditional mean embeddings (CME) operator and then present an operator stochastic approximation algorithm to learn the Koopman operator iteratively with control over the complexity of the representation. We provide both asymptotic and finite-time last-iterate guarantees of the online sparse learning algorithm with trajectory-based sampling with an analysis that is substantially more involved than that for finite-dimensional stochastic approximation. Numerical examples confirm the effectiveness of the proposed algorithm.