CONVERGENCE OF CONTINUOUS NORMALIZING FLOWS FOR LEARNING PROBABILITY DISTRIBUTIONS

作     者：Gao, By Yuan Huang, Jian Jiao, Yuling Zheng, Shurong 

作者机构：Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong School of Mathematics and Statistics Hubei Key Laboratory of Computational Science Wuhan University Wuhan China School of Mathematics and Statistics Northeast Normal University Changchun China 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Ordinary differential equations 

摘      要：Continuous normalizing flows (CNFs) are a generative method for learning probability distributions, which is based on ordinary differential equations. This method has shown remarkable empirical success across various applications, including large-scale image synthesis, protein structure prediction, and molecule generation. In this work, we study the theoretical properties of CNFs with linear interpolation in learning probability distributions from a finite random sample, using a flow matching objective function. We establish non-asymptotic error bounds for the distribution estimator based on CNFs, in terms of the Wasserstein-2 distance. The key assumption in our analysis is that the target distribution satisfies one of the following three conditions: it either has a bounded support, is strongly log-concave, or is a finite or infinite mixture of Gaussian distributions. We present a convergence analysis framework that encompasses the error due to velocity estimation, the discretization error, and the early stopping error. A key step in our analysis involves establishing the regularity properties of the velocity field and its estimator for CNFs constructed with linear interpolation. This necessitates the development of uniform error bounds with Lipschitz regularity control of deep ReLU networks that approximate the Lipschitz function class, which could be of independent interest. Our nonparametric convergence analysis offers theoretical guarantees for using CNFs to learn probability distributions from a finite random *** Codes 62G05, 68T07 © 2024, CC BY.