Denoising diffusion models are a novel class of generative algorithms that achieve state-of-the-art performance across a range of domains, including image generation and text-to-image tasks. Building on this success, diffusion models have recently been extended to the Riemannian manifold setting, broadening their applicability to a range of problems from the natural and engineering sciences. However, these Riemannian diffusion models are built on the assumption that their forward and backward processes are well-defined for all times, preventing them from being applied to an important set of tasks that consider manifolds defined via a set of inequality constraints. In this work, we introduce a principled framework to bridge this gap. We present two distinct noising processes based on (i) the logarithmic barrier metric and (ii) the reflected Brownian motion induced by the constraints. As existing diffusion model techniques cannot be applied in this setting, we derive new tools to define such models in our framework. We then demonstrate the practical utility of our methods on a number of synthetic and real-world tasks, including applications from robotics and protein design.

翻译：去噪扩散模型是一类新型生成算法，在图像生成和文本到图像任务等多个领域均取得了最先进的性能。基于这一成功，扩散模型最近被推广到黎曼流形设定，拓展了其在自然科学与工程科学中的一系列问题的适用性。然而，这些黎曼扩散模型建立在正向和反向过程对所有时间都定义良好的假设之上，这使其无法应用于一类重要任务——这些任务考虑通过一组不等式约束定义的流形。在本工作中，我们引入了一个严谨的框架来弥补这一不足。我们提出了两种不同的噪声化过程，分别基于：(i) 对数障碍度量；(ii) 由约束诱导的反射布朗运动。由于现有扩散模型技术无法应用于此设定，我们推导了新工具来定义此类模型。随后，我们在多个合成任务和现实任务（包括来自机器人学和蛋白质设计的应用）上展示了我们方法的实际效用。