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

TL;DR

This paper introduces KSWGD, a novel operator-theoretic generative modeling method that estimates spectral structures from data, ensuring linear convergence and improved sample quality across complex systems.

Contribution

It develops Koopman Spectral Wasserstein Gradient Descent, integrating Koopman theory with Wasserstein gradient descent for efficient, data-driven generative modeling without explicit potential knowledge.

Findings

Outperforms baselines in convergence speed

Achieves higher sample quality

Works on high-dimensional and complex systems

Abstract

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.