用谱方法统一生成模型，让训练更稳定高效。
Perron--Frobenius Operator Matching for Generative Modeling
- 基于佩龙-弗罗贝尼乌斯算子匹配密度演化路径
- 仅KL散度能保持目标等价性，简化为可计算的损失函数
- 加速训练与采样，适合高维生成任务
我们提出佩龙-弗罗贝尼乌斯算子匹配（PFOM），一种通过积分型佩龙-弗罗贝尼乌斯算子匹配密度演化过程的生成框架，可统一流模型、扩散模型和跳跃模型。我们证明在所有Bregman散度中，仅肯德尔-利布勒（KL）散度能保持密度级目标与样本条件目标之间的等价性，从而导出一个等价于科普曼路径匹配的实际损失。进一步开发了纳斯特罗夫加速的训练与采样方法，提升离散化稳定性并加速收敛。在高斯混合分布和双月数据集上，PFOM实现了更快的KL/W₂/MMD下降，且具有更高的实际运行效率。该方法将算子理论识别与现代生成建模相融合，为自适应字典与高维应用开辟新路径。
原文摘要 · Abstract (English)
We introduce Perron--Frobenius Operator Matching (PFOM), a generative framework that matches density evolution via the integral PF operator, subsuming flow, diffusion, and jump models. We prove that among Bregman divergences, only Kullback--Leibler divergence preserves equality between density-level and sample-conditioned objectives, yielding a practical loss equivalent to Koopman path matching. We further develop Nesterov-accelerated training and sampling that stabilize discretization and accelerate convergence. %On Gaussian mixtures and two-moons, PFOM achieves faster KL/$W_2$/MMD decrease and improved wall-clock efficiency with empirical validation. PFOM unifies operator-theoretic identification with modern generative modeling and opens paths to adaptive dictionaries and high-dimensional applications.
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