Diffusion Means and their Relation to Intrinsic and Extrinsic Means

In statistics on manifolds, we introduce a new family of location statistics describing centers of isotropic diffusion for different diffusion times. In contrast to the situation in Euclidean data, these diffusion means on manifolds do not generally coincide for different diffusion times. In the limit of vanishing diffusion time, diffusion means can be shown to converge to the intrinsic mean in general. For the limit of infinite diffusion time, we show for the circle and spheres of arbitrary dimension that diffusion means converge to the extrinsic mean in the canonical embedding. This yields an appealing interpretation of the extrinsic mean and a definition without reference to an embedding. Furthermore, we show that diffusion means with simultaneously estimated diffusion time have appealing regularity properties which can make them preferable to the intrinsic mean.