We present an estimate of the Wasserstein distance between the data distribution and the generation of score-based generative models, assuming an $\epsilon$-accurate approximation of the score and a Gaussian-type tail behavior of the data distribution. The complexity bound in dimension is $O(\sqrt{d})$, with a logarithmic constant. Such Gaussian tail assumption applies to the distribution of a compact support target with early stopping technique and the Bayesian posterior with a bounded observation operator. Corresponding convergence and complexity bounds are derived. The crux of the analysis lies in the Lipchitz bound of the score, which is related to the Hessian estimate of a viscous Hamilton-Jacobi equation (vHJ). This latter is demonstrated by employing a dimension independent kernel estimate. Consequently, our complexity bound scales linearly (up to a logarithmic constant) with the square root of the trace of the covariance operator, which relates to the invariant distribution of forward process. Our analysis also extends to the probabilistic flow ODE, as the sampling process.

翻译：本文针对基于分数的生成模型，在假设得分函数具有$\epsilon$级近似精度且数据分布呈现高斯型尾部行为的条件下，给出了数据分布与生成分布之间Wasserstein距离的估计。所得复杂度界在维度上为$O(\sqrt{d})$，并带有对数常数。此类高斯尾部假设适用于采用早停技术的紧支撑目标分布，以及具有有界观测算子的贝叶斯后验分布。我们推导了相应的收敛性与复杂度界。分析的关键在于得分函数的Lipschitz界，该界与粘性Hamilton-Jacobi方程（vHJ）的Hessian估计相关。后者通过采用与维度无关的核估计得以证明。因此，我们的复杂度界与协方差算子迹的平方根呈线性比例关系（直至一个对数常数），该协方差算子与正向过程的不变分布相关。我们的分析同样可推广至作为采样过程的概率流常微分方程。