Nonparametric Sparse Online Learning of the Koopman Operator

2025-01-27

AI Summary

Koopman operator learning for nonlinear dynamical systems traditionally relies on pre-specified, finite-dimensional observable spaces, imposing restrictive closure assumptions. Method: This paper introduces an online sparse learning paradigm within a reproducing kernel Hilbert space (RKHS), establishing— for the first time—theoretical equivalence between the Koopman operator and the conditional mean embedding operator. The proposed framework is fully nonparametric and does not assume observables’ closure. It integrates conditional mean embeddings, stochastic operator approximation, sparse kernel methods, and trajectory-driven optimization. Contribution/Results: We provide the first finite-time last-iterate convergence guarantee under trajectory sampling. Numerical experiments demonstrate substantial improvements in dynamic prediction accuracy and generalization performance, while maintaining representation sparsity and robust convergence—empirically validating the theoretical guarantees.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning

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.

Problem

• Koopman Operator
• Adaptive Learning
• Complex System Prediction

Innovation

• Koopman operator
• reproducing kernel Hilbert spaces
• conditional mean embeddings