# Perron--Frobenius Operator Matching for Generative Modeling

arXiv: https://arxiv.org/abs/2606.17465 · HTML: https://mlsanity.com/paper/2606.17465/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Verification of Image-based Neural Network Controllers Using Generative Models](https://mlsanity.com/paper/2105.07091/) (2021)
- [Data-Driven Approximation of the Perron-Frobenius Operator Using the Wasserstein Metric](https://mlsanity.com/paper/2011.00759/) (2020)
- [GenControl: Generative AI-Driven Autonomous Design of Control Algorithms](https://mlsanity.com/paper/2506.12554/) (2025)

## Abstract (arXiv)

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/W2/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.