Sampling Hypersurfaces through Diffusion

We are interested in efficient algorithms for generating random samples from geometric objects such as Riemannian manifolds. As a step in this direction, we consider the problem of generating random samples from smooth hypersurfaces that may be represented as the boundary ∂A of a domain A ⊂ R of Euclidean space. A is specified through a membership oracle and we assume access to a blackbox that can generate uniform random samples from A. By simulating a diffusion process with a suitably chosen time constant t, we are able to construct algorithms that can generate points (approximately) on ∂A according to a (approximately) uniform distribution. We have two classes of related but distinct results. First, we consider A to be a convex body whose boundary is the union of finitely many smooth pieces, and provide an algorithm (Csample) that generates (almost) uniformly random points from the surface of this body, and prove that its complexity is O∗( d 4 2 ) per sample, where 2 is the variation distance. Next, we consider A to be a potentially non-convex body whose boundary is a smooth (co-dimension one) manifold with a bound on its absolute curvature and diameter. We provide an algorithm (Msample) that generates almost uniformly random points from ∂A, and prove that its complexity is O( R √ 2τ ) where 1 τ is a bound on the curvature of ∂A, and R is the radius of a circumscribed ball.