Generative Modeling through Spectral Analysis of Koopman Operator
Yuanchao Xu, Fengyi Li, Masahiro Fujisawa, Youssef Marzouk, Isao Ishikawa
TL;DR
The paper tackles the challenge of generative modeling without access to explicit target densities by proposing KSWGD, a training-free method that marries Koopman spectral analysis with Wasserstein gradient flows. By estimating a finite-rank surrogate for the inverse Langevin generator from trajectory data, KSWGD achieves fast, linearly convergent sampling and provides explicit error bounds tied to spectral truncation and data quality. The framework is grounded in a Feynman-Kac interpretation and validated across diverse tasks, including manifold sampling, metastable systems, MNIST latent generation, and SPDEs, often outperforming data-driven baselines that rely on neural networks. The work offers a scalable alternative to neural training for high-quality sample generation and opens avenues for conditional sampling via nonzero killing rates.
Abstract
We propose Koopman Spectral Wasserstein Gradient Descent (KSWGD), a generative modeling framework that combines operator-theoretic spectral analysis with optimal transport. The novel insight is that the spectral structure required for accelerated Wasserstein gradient descent can be directly estimated from trajectory data via Koopman operator approximation which can eliminate the need for explicit knowledge of the target potential or neural network training. We provide rigorous convergence analysis and establish connection to Feynman-Kac theory that clarifies the method's probabilistic foundation. Experiments across diverse settings, including compact manifold sampling, metastable multi-well systems, image generation, and high dimensional stochastic partial differential equation, demonstrate that KSWGD consistently achieves faster convergence than other existing methods while maintaining high sample quality.