Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling on image domains. In addition, their extension to Riemannian manifolds has facilitated a range of applications across the natural sciences. While many of these problems stand to benefit from the ability to specify arbitrary, domain-informed constraints, this setting is not covered by the existing (Riemannian) diffusion model methodology. Recent work has attempted to address this issue by constructing novel noising processes based on the reflected Brownian motion and logarithmic barrier methods. However, the associated samplers are either computationally burdensome or only apply to convex subsets of Euclidean space. 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：去噪扩散模型最近已成为图像领域生成建模的主流范式。此外，将其扩展至黎曼流形促进了自然科学领域的一系列应用。虽然许多此类问题有望受益于指定任意领域约束的能力，但现有（黎曼）扩散模型方法尚未涵盖这一场景。近期研究通过基于反射布朗运动和对数障碍方法构造新型加噪过程来尝试解决该问题，然而相关采样器要么计算负担过重，要么仅适用于欧几里得空间的凸子集。本文提出一种基于Metropolis采样的替代性简易加噪方案，与先前采样器相比，该方法在计算效率和实证性能方面均实现了显著提升。具有独立意义的是，我们证明了该新过程等价于反射布朗运动的有效离散化形式。我们通过一系列包含凸约束与非凸约束的问题场景（包括地理空间建模、机器人学和蛋白质设计等应用）展示了该方法的可扩展性与灵活性。