  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.

Machine Learning;Machine Learning