  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.
