Generative Modeling through Spectral Analysis of Koopman Operator

2025-12-21

AI Summary

This work addresses the limitations of Wasserstein gradient descent in generative modeling—namely, its reliance on prior knowledge of the target potential function or costly neural network training. We propose the Koopman Spectral Wasserstein Gradient Descent (KSWGD) framework, the first to deeply integrate Koopman operator spectral analysis with optimal transport. KSWGD directly estimates the spectral structure from trajectory data, enabling potential-free and training-free generative dynamical modeling. Theoretically, we establish rigorous connections between KSWGD and the Feynman–Kac formula as well as solutions to stochastic partial differential equations (SPDEs). Algorithmically, the method evolves the Wasserstein gradient flow via Koopman spectral approximation. Experiments demonstrate that KSWGD achieves significantly faster convergence than state-of-the-art methods on manifold sampling, multiscale/multistable system simulation, image generation, and high-dimensional SPDE solving—while preserving sample quality.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• Accelerates Wasserstein gradient descent via Koopman spectral analysis
• Estimates spectral structure from trajectory data without neural training
• Applies to manifold sampling, metastable systems, and image generation

Innovation

• Combines operator spectral analysis with optimal transport
• Estimates spectral structure from data via Koopman operator
• Achieves faster convergence without neural network training