用随机微分方程统一解释生成模型的变分原理。
Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective
- 从变分下界出发，统一框架解释扩散、得分匹配和流匹配方法。
- 通过一维密度建模对比不同参数化方案的生成效果。
- 适合想理解生成模型底层机制的研究者入门。
随机微分方程在生成建模中取得显著进展，广泛应用于图像、视频和生物分子生成。本文提供了一个自包含且非正式的介绍，涵盖微分方程、概率框架及描述变量边缘分布演化的福克-普朗克方程。推导了对数似然的变分下界（证据下界，ELBO），并以此为起点讨论扩散模型、得分匹配与流匹配。这些方法均可视为最一般变分方法的具体参数化形式。通过一维密度建模问题，对比不同参数化方案的性能表现。
原文摘要 · Abstract (English)
The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.
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