Stochastic Heat Kernel Estimation on Sampled Manifolds

Abstract

The heat kernel is a fundamental geometric object associated to every Riemannian manifold, used across applications in computer vision, graphics, and machine learning. In this article, we propose a novel computational approach to estimating the heat kernel of a statistically sampled manifold (e.g. meshes or point clouds), using its representation as the transition density function of Brownian motion on the manifold. Our approach first constructs a set of local approximations to the manifold via moving least squares. We then simulate Brownian motion on the manifold by stochastic numerical integration of the associated Ito diffusion system. By accumulating a number of these trajectories, a kernel density estimation method can then be used to approximate the transition density function of the diffusion process, which is equivalent to the heat kernel. We analyse our algorithm on the 2-sphere, as well as on shapes in 3D. Our approach is readily parallelizable and can handle manifold samples of large size as well as surfaces of high co-dimension, since all the computations are local. We relate our method to the standard approaches in diffusion geometry and discuss directions for future work.