The efficacy of modern generative models is commonly contingent upon the precision of score estimation along the diffusion path, with a focus on diffusion models and their ability to generate high-quality data samples. This study delves into the application of reverse diffusion to Monte Carlo sampling. It is shown that score estimation can be transformed into a mean estimation problem via the decomposition of the transition kernel. By estimating the mean of the posterior distribution, we derive a novel Monte Carlo sampling algorithm from the reverse diffusion process, which is distinct from traditional Markov Chain Monte Carlo (MCMC) methods. We calculate the error requirements and sample size for the posterior distribution, and use the result to derive an algorithm that can approximate the target distribution to any desired accuracy. Additionally, by estimating the log-Sobolev constant of the posterior distribution, we show under suitable conditions the problem of sampling from the posterior can be easier than direct sampling from the target distribution using traditional MCMC techniques. For Gaussian mixture models, we demonstrate that the new algorithm achieves significant improvement over the traditional Langevin-style MCMC sampling methods both theoretically and practically. Our algorithm offers a new perspective and solution beyond classical MCMC algorithms for challenging complex distributions.

翻译：现代生成模型（尤其是扩散模型）的效能通常依赖于扩散路径上分数估计的精度，其核心能力体现在生成高质量数据样本。本研究深入探索了反向扩散在蒙特卡洛采样中的应用。研究表明，通过分解转移核，分数估计可转化为均值估计问题。基于对后验分布均值的估计，我们从反向扩散过程中推导出一种不同于传统马尔可夫链蒙特卡洛（MCMC）方法的新型蒙特卡洛采样算法。我们计算了后验分布的误差要求与样本量，并利用该结果推导出能以任意精度逼近目标分布的算法。此外，通过估计后验分布的对数Sobolev常数，我们证明在适当条件下，从后验分布采样可能比使用传统MCMC方法直接从目标分布采样更容易。针对高斯混合模型，我们的新算法在理论与实践中均显著优于传统朗之万式MCMC采样方法。该算法为处理复杂分布问题提供了超越经典MCMC算法的新视角与解决方案。