Diffusion models have become the leading distribution-learning method in
recent years. Herein, we introduce structure-preserving diffusion processes, a
family of diffusion processes for learning distributions that possess
additional structure, such as group symmetries, by developing theoretical
conditions under which the diffusion transition steps preserve said symmetry.
While also enabling equivariant data sampling trajectories, we exemplify these
results by developing a collection of different symmetry equivariant diffusion
models capable of learning distributions that are inherently symmetric.
Empirical studies, over both synthetic and real-world datasets, are used to
validate the developed models adhere to the proposed theory and are capable of
achieving improved performance over existing methods in terms of sample
equality. We also show how the proposed models can be used to achieve
theoretically guaranteed equivariant image noise reduction without prior
knowledge of the image orientation.

最近几年扩散模型已成为主要的分布学习方法。本文介绍了一种称为结构保持扩散过程的扩散模型系列，用于学习具有额外结构特征（例如群对称性）的分布。该研究通过开发理论条件，使扩散转移步骤保持所述对称性，从而使得数据采样轨迹具有等变性。我们以开发一系列不同对称等变的扩散模型来验证这些结果，并能够学习具有固有对称性的分布。通过在合成和真实数据集上进行实证研究，我们验证了所提出的模型符合提出的理论，并在样本均等性方面能够实现比现有方法更好的性能。我们还展示了如何使用所提出的模型，在不事先了解图像方向的情况下实现理论上有保证的等变图像降噪。