Functional Flow Matching
Gavin Kerrigan, Giosue Migliorini, Padhraic Smyth
Abstract
We propose Functional Flow Matching (FFM), a function-space generative model that generalizes the recently-introduced Flow Matching model to operate directly in infinite-dimensional spaces. Our approach works by first defining a path of probability measures that interpolates between a fixed Gaussian measure and the data distribution, followed by learning a vector field on the underlying space of functions that generates this path of measures. Our method does not rely on likelihoods or simulations, making it well-suited to the function space setting. We provide both a theoretical framework for building such models and an empirical evaluation of our techniques. We demonstrate through experiments on synthetic and real-world benchmarks that our proposed FFM method outperforms several recently proposed function-space generative models.
BibTeX
@InProceedings{pmlr-v238-kerrigan24a,
title = {Functional Flow Matching},
author = {Kerrigan, Gavin and Migliorini, Giosue and Smyth, Padhraic},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {3934--3942},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/kerrigan24a/kerrigan24a.pdf},
url = {https://proceedings.mlr.press/v238/kerrigan24a.html},
abstract = {We propose Functional Flow Matching (FFM), a function-space generative model that generalizes the recently-introduced Flow Matching model to operate directly in infinite-dimensional spaces. Our approach works by first defining a path of probability measures that interpolates between a fixed Gaussian measure and the data distribution, followed by learning a vector field on the underlying space of functions that generates this path of measures. Our method does not rely on likelihoods or simulations, making it well-suited to the function space setting. We provide both a theoretical framework for building such models and an empirical evaluation of our techniques. We demonstrate through experiments on synthetic and real-world benchmarks that our proposed FFM method outperforms several recently proposed function-space generative models.}
}