用于学习概率分布的连续归一化流的收敛性
Convergence of Continuous Normalizing Flows for Learning Probability Distributions
AI总结：
本文从理论上研究了基于线性插值和流匹配目标函数的连续归一化流从有限样本学习概率分布的收敛性，在目标分布具有有界支撑、强对数凹或为高斯混合的条件下，建立了Wasserstein-2距离的非渐近误差界。
摘要：
连续归一化流（CNFs）是一种基于常微分方程的学习概率分布的生成方法。该方法在包括大规模图像合成、蛋白质结构预测和分子生成在内的各种应用中取得了显著的实证成功。在本工作中，我们使用流匹配目标函数，研究具有线性插值的CNFs从有限随机样本中学习概率分布的理论性质。我们以Wasserstein-2距离为度量，为基于CNFs的分布估计器建立了非渐近误差界。我们分析中的关键假设是目标分布满足以下三个条件之一：具有有界支撑、是强对数凹的，或者是高斯分布的有限或无限混合。我们提出了一个收敛分析框架，该框架涵盖了速度估计误差、离散化误差和早停误差。我们分析中的一个关键步骤是建立为线性插值CNFs构造的速度场及其估计器的正则性性质。这需要开发逼近Lipschitz函数类的深度ReLU网络的一致误差界及其Lipschitz正则性控制，该结果可能具有独立的研究意义。我们的非参数收敛分析为使用CNFs从有限随机样本中学习概率分布提供了理论保证。
英文摘要：
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 sample.
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