用谱方法建模生成分布，解决梯度消失问题，提升采样速度与质量。
Generative Modeling through Koopman Spectral Analysis: An Operator-Theoretic Perspective
- 基于柯普曼算子从轨迹数据直接估计分布谱结构
- 保持近似恒定耗散率，实现线性收敛且避免梯度消失
- 适合高维复杂系统生成建模，尤其多阱势能系统
我们提出柯普曼谱沃瑟斯坦梯度下降（KSWGD），一种基于粒子的生成建模框架，通过柯普曼理论学习朗之万生成器，并结合沃瑟斯坦梯度下降。核心洞察在于，目标分布的谱结构可直接从轨迹数据中通过柯普曼算子估计，无需显式知道目标势能函数。此外，我们证明KSWGD维持近似恒定的耗散率，从而建立线性收敛性，克服了现有核基粒子方法中的梯度消失问题。我们进一步提供费曼-卡克解释，阐明该方法的概率基础。在紧致流形、具有代谢能垒的多阱系统以及高维随机偏微分方程上的实验表明，KSWGD在收敛速度和样本质量上均持续优于基线方法。
原文摘要 · Abstract (English)
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.
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