Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling. Their extension to Riemannian manifolds has facilitated their application to an array of problems in the natural sciences. Yet, in many practical settings, such manifolds are defined by a set of constraints and are not covered by the existing (Riemannian) diffusion model methodology. Recent work has attempted to address this issue by employing novel noising processes based on logarithmic barrier methods or reflected Brownian motions. However, the associated samplers are computationally burdensome as the complexity of the constraints increases. In this paper, we introduce an alternative simple noising scheme based on Metropolis sampling that affords substantial gains in computational efficiency and empirical performance compared to the earlier samplers. Of independent interest, we prove that this new process corresponds to a valid discretisation of the reflected Brownian motion. We demonstrate the scalability and flexibility of our approach on a range of problem settings with convex and non-convex constraints, including applications from geospatial modelling, robotics and protein design.

翻译：去噪扩散模型近来已成为生成建模的主流范式。将其推广到黎曼流形上，促进了其在自然科学中一系列问题中的应用。然而，在许多实际场景中，这类流形由一组约束条件定义，且现有（黎曼）扩散模型方法体系无法覆盖。近期研究尝试通过采用基于对数障碍方法或反射布朗运动的新型加噪过程来解决此问题。然而，随着约束复杂度的增加，相关采样器的计算负担日益沉重。本文提出了一种基于Metropolis采样的替代性简易加噪方案，与先前的采样器相比，该方案在计算效率和经验性能上均实现了显著提升。独立而言，我们证明了这一新过程对应于反射布朗运动的一种有效离散化形式。我们在一系列具有凸约束与非凸约束的问题场景中（包括来自地理空间建模、机器人学和蛋白质设计的应用）展示了该方法的可扩展性与灵活性。