  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.
