This paper attempts to present the stochastic differential equations of diffusion models in a manner that is accessible to a broad audience. The diffusion process is defined over a population density in R^d. Of particular interest is a population of images. In a diffusion model one first defines a diffusion process that takes a sample from the population and gradually adds noise until only noise remains. The fundamental idea is to sample from the population by a reverse-diffusion process mapping pure noise to a population sample. The diffusion process is defined independent of any ``interpretation'' but can be analyzed using the mathematics of variational auto-encoders (the ``VAE interpretation'') or the Fokker-Planck equation (the ``score-matching intgerpretation''). Both analyses yield reverse-diffusion methods involving the score function. The Fokker-Planck analysis yields a family of reverse-diffusion SDEs parameterized by any desired level of reverse-diffusion noise including zero (deterministic reverse-diffusion). The VAE analysis yields the reverse-diffusion SDE at the same noise level as the diffusion SDE. The VAE analysis also yields a useful expression for computing the population probabilities of a given point (image). This formula for the probability of a given point does not seem to follow naturally from the Fokker-Planck analysis. Much, but apparently not all, of the mathematics presented here can be found in the literature. Attributions are given at the end of the paper.

该论文通过高斯分布得出扩散模型的微分方程和似然函数，分析了前向和后向随机微分方程及似然函数的变分自编码器和分数匹配方法，得到了反扩散常微分方程和基于噪声级别参数化的一族反扩散随机微分方程。

扩散模型的数学