There is a long history, as well as a recent explosion of interest, in statistical and generative modeling approaches based on score functions -- derivatives of the log-likelihood of a distribution. In seminal works, Hyv\"arinen proposed vanilla score matching as a way to learn distributions from data by computing an estimate of the score function of the underlying ground truth, and established connections between this method and established techniques like Contrastive Divergence and Pseudolikelihood estimation. It is by now well-known that vanilla score matching has significant difficulties learning multimodal distributions. Although there are various ways to overcome this difficulty, the following question has remained unanswered -- is there a natural way to sample multimodal distributions using just the vanilla score? Inspired by a long line of related experimental works, we prove that the Langevin diffusion with early stopping, initialized at the empirical distribution, and run on a score function estimated from data successfully generates natural multimodal distributions (mixtures of log-concave distributions).

翻译：在统计与生成建模中，基于得分函数（即分布对数似然的导数）的方法有着悠久的历史，并且近期引发了研究热潮。在开创性工作中，Hyvärinen提出朴素得分匹配方法，通过计算底层真实数据得分函数的估计值来从数据中学习分布，并建立了该方法与对比散度、伪似然估计等经典技术之间的联系。目前学界普遍认识到，朴素得分匹配在学习多模态分布时存在显著困难。尽管已有多种方法可克服这一难题，但以下问题仍未得到解答——是否存在仅利用朴素得分函数对多模态分布进行自然采样的方法？受一系列相关实验工作的启发，我们证明：以经验分布为初始分布、采用早停策略的朗之万扩散方法，在基于数据估计的得分函数上运行，能够成功生成自然的多模态分布（对数凹分布的混合）。