# Generative Modeling through Koopman Spectral Analysis: An Operator-Theoretic Perspective

arXiv: https://arxiv.org/abs/2512.18837 · HTML: https://mlsanity.com/paper/2512.18837/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 46 references that are in the map.
- First posted on arXiv 21 Dec 2025.

## Builds on (most cited first, 14 of 14)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [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)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
- [Diffusion map particle systems for generative modeling](https://mlsanity.com/paper/2304.00200/) (2023)

## Abstract (arXiv)

We propose Koopman Spectral Wasserstein Gradient Descent (KSWGD), a particle-based generative modeling framework that learns the Langevin generator via Koopman theory and integrates it with Wasserstein gradient descent. Our key insight is that this spectral structure of the underlying distribution can be directly estimated from trajectory data via the Koopman operator, eliminating the need for explicit knowledge of the target potential. Additionally, we prove that KSWGD maintains an approximately constant dissipation rate, thereby establishing linear convergence and overcoming the vanishing-gradient phenomenon that hinders existing kernel-based particle methods. We further provide a Feynman--Kac interpretation that clarifies the method's probabilistic foundation. Experiments on compact manifolds, metastable multi-well systems, and high-dimensional stochastic partial differential equations demonstrate that KSWGD consistently outperforms baselines in both convergence speed and sample quality.