Denoising diffusion models are a recent class of generative models which achieve state-of-the-art results in many domains such as unconditional image generation and text-to-speech tasks. They consist of a noising process destroying the data and a backward stage defined as the time-reversal of the noising diffusion. Building on their success, diffusion models have recently been extended to the Riemannian manifold setting. Yet, these Riemannian diffusion models require geodesics to be defined for all times. While this setting encompasses many important applications, it does not include manifolds defined via a set of inequality constraints, which are ubiquitous in many scientific domains such as robotics and protein design. In this work, we introduce two methods to bridge this gap. First, we design a noising process based on the logarithmic barrier metric induced by the inequality constraints. Second, we introduce a noising process based on the reflected Brownian motion. As existing diffusion model techniques cannot be applied in this setting, we derive new tools to define such models in our framework. We empirically demonstrate the applicability of our methods to a number of synthetic and real-world tasks, including the constrained conformational modelling of protein backbones and robotic arms.

翻译：去噪扩散模型是近期一类生成模型，在无条件图像生成和文本转语音等许多任务中取得了最先进的成果。它们由一个破坏数据的加噪过程和一个定义为加噪扩散时间反转的反向阶段组成。基于其成功，扩散模型最近被扩展到黎曼流形设定中。然而，这些黎曼扩散模型要求测地线在所有时间均有定义。虽然这一设定涵盖了许多重要应用，但它并不包括由一组不等式约束定义的流形，而这些约束在机器人和蛋白质设计等众多科学领域中普遍存在。在这项工作中，我们引入了两种方法来弥补这一差距。首先，我们设计了一个基于不等式约束诱导的对数障碍度量的加噪过程。其次，我们引入了一个基于反射布朗运动的加噪过程。由于现有的扩散模型技术无法应用于此设定，我们推导了新工具来在我们的框架中定义此类模型。我们通过实验证明了我们的方法在多个合成和真实世界任务中的适用性，包括蛋白质骨架和机械臂的约束构象建模。