  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.
