Diffusion models are typically trained using score matching, yet score matching is agnostic to the particular forward process that defines the model. This paper argues that Markov diffusion models enjoy an advantage over other types of diffusion model, as their associated operators can be exploited to improve the training process. In particular, (i) there exists an explicit formal solution to the forward process as a sequence of time-dependent kernel mean embeddings; and (ii) the derivation of score-matching and related estimators can be streamlined. Building upon (i), we propose Riemannian diffusion kernel smoothing, which ameliorates the need for neural score approximation, at least in the low-dimensional context; Building upon (ii), we propose operator-informed score matching, a variance reduction technique that is straightforward to implement in both low- and high-dimensional diffusion modeling and is demonstrated to improve score matching in an empirical proof-of-concept.

翻译：扩散模型通常使用分数匹配进行训练，然而分数匹配与定义模型的具体前向过程无关。本文认为，马尔可夫扩散模型相较于其他类型的扩散模型具有优势，因为可以利用其关联的算子来改进训练过程。具体而言：(i) 前向过程存在一个显式形式解，表现为一系列时间依赖的核均值嵌入；(ii) 分数匹配及相关估计量的推导可以得到简化。基于(i)，我们提出了黎曼扩散核平滑方法，该方法至少在低维情境下改善了对神经分数近似的需求；基于(ii)，我们提出了算子信息分数匹配，这是一种方差缩减技术，在低维和高维扩散建模中均易于实现，并通过经验性概念验证证明了其能改进分数匹配。