# Avoiding spectral pollution for transfer operators using residuals

arXiv: https://arxiv.org/abs/2507.16915 · HTML: https://mlsanity.com/paper/2507.16915/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 78 references that are in the map.
- First posted on arXiv 22 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Deep Learning Enhanced Dynamic Mode Decomposition](https://mlsanity.com/paper/2108.04433/) (2021)
- [Adversarial dynamical systems characterize when data-driven learning succeeds or fails](https://mlsanity.com/paper/2407.06312/) (2024)
- [A kernel-based approach to molecular conformation analysis](https://mlsanity.com/paper/1809.11092/) (2018)
- [Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size](https://mlsanity.com/paper/2403.05891/) (2024)
- [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024)
- [Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2506.15782/) (2025)
- [Transition path theory for Langevin dynamics on manifold: optimal control and data-driven solver](https://mlsanity.com/paper/2010.09988/) (2020)
- [Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos](https://mlsanity.com/paper/2305.08074/) (2023)
- [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024)

## Abstract (arXiv)

Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.