# On the Convergence of Hermitian Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2401.03192 · HTML: https://mlsanity.com/paper/2401.03192/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 6 Jan 2024.

## Builds on (most cited first, 9 of 9)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems](https://mlsanity.com/paper/2308.10697/) (2023)

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

- [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024)
- [Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning](https://mlsanity.com/paper/2606.05131/) (2026)

## Abstract (arXiv)

We study the convergence of Hermitian Dynamic Mode Decomposition (DMD) to the spectral properties of self-adjoint Koopman operators. Hermitian DMD is a data-driven method that approximates the Koopman operator associated with an unknown nonlinear dynamical system, using discrete-time snapshots. This approach preserves the self-adjointness of the operator in its finite-dimensional approximations. revWe prove that, under suitably broad conditions, the spectral measures corresponding to the eigenvalues and eigenfunctions computed by Hermitian DMD converge to those of the underlying Koopman operator. This result also applies to skew-Hermitian systems (after multiplication by i), applicable to generators of continuous-time measure-preserving systems. Along the way, we establish a general theorem on the convergence of spectral measures for finite sections of self-adjoint operators, including those that are unbounded, which is of independent interest to the wider spectral community. We numerically demonstrate our results by applying them to two-dimensional Schr"odinger equations.