Structure Preserving Diffusion Models
This work addresses the challenge of preserving geometric structures in diffusion models for applications like medical imaging, representing an incremental improvement by complementing existing conditions and proposing a new framework.
The paper tackled the problem of learning distributions with inherent structures, such as group symmetries, by developing structure-preserving diffusion models, resulting in models that maintain equivariance and achieve high sample quality in experiments on real-world medical images.
In recent years, diffusion models have become the leading approach for distribution learning. This paper focuses on structure-preserving diffusion models (SPDM), a specific subset of diffusion processes tailored for distributions with inherent structures, such as group symmetries. We complement existing sufficient conditions for constructing SPDMs by proving complementary necessary ones. Additionally, we propose a new framework that considers the geometric structures affecting the diffusion process. Leveraging this framework, we design a structure-preserving bridge model that maintains alignment between the model's endpoint couplings. Empirical evaluations on equivariant diffusion models demonstrate their effectiveness in learning symmetric distributions and modeling transitions between them. Experiments on real-world medical images confirm that our models preserve equivariance while maintaining high sample quality. We also showcase the practical utility of our framework by implementing an equivariant denoising diffusion bridge model, which achieves reliable equivariant image noise reduction and style transfer, irrespective of prior knowledge of image orientation.