Analysis of Learning a Flow-based Generative Model from Limited Sample Complexity
Hugo Cui, Florent Krzakala, Eric Vanden-Eijnden, Lenka Zdeborova
Abstract
We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number $n$ of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the means of the generated mixture and the mean of the target mixture, which we show decays as $\Theta_n(\frac{1}{n})$. Finally, this rate is shown to be in fact Bayes-optimal.
BibTeX
@inproceedings{
cui2024analysis,
title={Analysis of Learning a Flow-based Generative Model from Limited Sample Complexity},
author={Hugo Cui and Florent Krzakala and Eric Vanden-Eijnden and Lenka Zdeborova},
booktitle={The Twelfth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=ndCJeysCPe}
}