# Nonparametric Sparse Online Learning of the Koopman Operator

arXiv: https://arxiv.org/abs/2501.16489 · HTML: https://mlsanity.com/paper/2501.16489/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 58 references that are in the map.
- First posted on arXiv 27 Jan 2025.

## Builds on (most cited first, 20 of 20)

- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning](https://mlsanity.com/paper/1902.00923/) (2019)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning](https://mlsanity.com/paper/1905.11425/) (2019)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [A Consistent Regularization Approach for Structured Prediction](https://mlsanity.com/paper/1605.07588/) (2016)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [Online dynamic mode decomposition for time-varying systems](https://mlsanity.com/paper/1707.02876/) (2017)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Nonparametric approximation of conditional expectation operators](https://mlsanity.com/paper/2012.12917/) (2020)
- [Learning to Forecast Dynamical Systems from Streaming Data](https://mlsanity.com/paper/2109.09703/) (2021)
- [Parsimonious Online Learning with Kernels via Sparse Projections in Function Space](https://mlsanity.com/paper/1612.04111/) (2016)
- [Adversarial dynamical systems characterize when data-driven learning succeeds or fails](https://mlsanity.com/paper/2407.06312/) (2024)
- [Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability](https://mlsanity.com/paper/2412.02811/) (2024)

## Abstract (arXiv)

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.