# Representer Theorem for Learning Koopman Operators

arXiv: https://arxiv.org/abs/2208.01681 · HTML: https://mlsanity.com/paper/2208.01681/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2022.

## Built on by (most cited first, 9 of 9)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting](https://mlsanity.com/paper/2506.14113/) (2025)
- [Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control](https://mlsanity.com/paper/2602.02592/) (2026)
- [Operator-Theoretic Generalization Bounds for Multitask Deep Learning](https://mlsanity.com/paper/2608.15982/) (2026)
- [DeepKoopFormer: A Koopman Enhanced Transformer Based Architecture for Time Series Forecasting](https://mlsanity.com/paper/2508.02616/) (2025)
- [Learning to control switching nonlinear systems with Koopman operator regression](https://mlsanity.com/paper/2607.11344/) (2026)
- [Sample-Weighted End-to-End Trace-Norm Geometry for Multitask Learning](https://mlsanity.com/paper/2609.29520/) (2026)

## Abstract (arXiv)

In this work, the problem of learning Koopman operator of a discrete-time autonomous system is considered. The learning problem is formulated as a constrained regularized empirical loss minimization in the infinite-dimensional space of linear operators. We show that under certain but general conditions, a representer theorem holds for the learning problem. This allows reformulating the problem in a finite-dimensional space without any approximation and loss of precision. Following this, we consider various cases of regularization and constraints in the learning problem, including the operator norm, the Frobenius norm, rank, nuclear norm, and stability. Subsequently, we derive the corresponding finite-dimensional problem. Furthermore, we discuss the connection between the proposed formulation and the extended dynamic mode decomposition. Finally, we provide an illustrative numerical example.