Perron--Frobenius Operator Matching for Generative Modeling

2026-06-15

AI Summary

This work proposes a unified framework for modeling the density evolution of diverse generative processes—including flows, diffusions, and jumps—by matching the dynamics governed by the Perron–Frobenius operator. The approach enforces consistency between the generated density and the conditional target distribution through Kullback–Leibler (KL) divergence, while leveraging Koopman path matching to enable efficient training. It is the first to apply Perron–Frobenius operator matching to generative modeling, theoretically establishing the uniqueness of KL divergence in ensuring consistency between densities and samples, and thereby bridging operator-theoretic principles with modern generative models. Experiments on Gaussian mixture and two-moons datasets demonstrate that the method significantly accelerates convergence in terms of KL divergence, Wasserstein-2 distance, and maximum mean discrepancy, while improving wall-clock efficiency during both training and sampling.

Abstract

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.

Problem

• generative modeling
• density evolution
• Perron--Frobenius operator
• Bregman divergences
• Kullback--Leibler divergence

Innovation

• Perron–Frobenius operator
• generative modeling
• Kullback–Leibler divergence
• Koopman matching
• Nesterov acceleration