Perron--Frobenius Operator Matching for Generative Modeling
For generative modeling practitioners, PFOM provides a unified theoretical framework and practical training/sampling acceleration, though results are shown only on simple 2D benchmarks.
Perron-Frobenius Operator Matching (PFOM) is introduced as a generative framework unifying flow, diffusion, and jump models via operator matching. It achieves faster convergence in KL, W2, and MMD on Gaussian mixtures and two-moons, with improved wall-clock efficiency.
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.