Generative Modeling through Spectral Analysis of Koopman Operator
This addresses generative modeling for applications like image generation and high-dimensional systems, offering a novel approach that eliminates the need for explicit target potential knowledge or neural network training, though it appears incremental as it builds on existing operator-theoretic and optimal transport methods.
The paper tackled generative modeling by proposing Koopman Spectral Wasserstein Gradient Descent (KSWGD), which uses spectral analysis of the Koopman operator to accelerate Wasserstein gradient descent, resulting in faster convergence than existing methods across diverse settings while maintaining high sample quality.
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.