# Nonparametric Sparse Online Learning of the Koopman Operator

arXiv: https://arxiv.org/abs/2405.07432 · HTML: https://mlsanity.com/paper/2405.07432/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 58 references that are in the map.
- First posted on arXiv 13 May 2024.

## Builds on (most cited first, 18 of 18)

- [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)

## Built on by (most cited first, 2 of 2)

- [Learning Ergodic Dynamical Systems from a Finite Trajectory](https://mlsanity.com/paper/2607.22399/) (2026)
- [Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators](https://mlsanity.com/paper/2608.06155/) (2026)

## Abstract (arXiv)

The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. However, existing data-driven approaches to learning the Koopman operator rely on batch data. In this work, we present a sparse online learning algorithm that learns the Koopman operator iteratively via stochastic approximation, with explicit control over model complexity and provable convergence guarantees. Specifically, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and address the mis-specified scenario where the dynamics may escape the chosen RKHS. In this mis-specified setting, we relate the Koopman operator to the conditional mean embeddings (CME) operator. We further establish both asymptotic and finite-time convergence guarantees for our learning algorithm in mis-specified setting, with trajectory-based sampling where the data arrive sequentially over time. Numerical experiments demonstrate the algorithm's capability to learn unknown nonlinear dynamics.