The Long Time Limit of Diffusion Means

TL;DR

This paper investigates the long-time behavior of diffusion means on manifolds, showing convergence to extrinsic means on spheres and extending results to real projective spaces, with conjectures for broader classes of spaces.

Contribution

It extends the understanding of diffusion means' limits to real projective spaces and conjectures a general behavior for symmetric spaces with isometric embeddings.

Findings

Diffusion means approach intrinsic means for short times.

On spheres, diffusion means converge to extrinsic means for long times.

Extension of convergence results to real projective spaces.

Abstract

In statistics on manifolds, the notion of the mean of a probability distribution becomes more involved than in a linear space. Several location statistics have been proposed, which reduce to the ordinary mean in Euclidean space. A relatively new family of contenders in this field are Diffusion Means, which are a one parameter family of location statistics modeled as initial points of isotropic diffusion with the diffusion time as parameter. It is natural to consider limit cases of the diffusion time parameter and it turns out that for short times the diffusion mean set approaches the intrinsic mean set. For long diffusion times, the limit is less obvious but for spheres of arbitrary dimension the diffusion mean set has been shown to converge to the extrinsic mean set. Here, we extend this result to the real projective spaces in their unique smooth isometric embedding into a linear space. We conjecture that the long time limit is always given by the extrinsic mean in the isometric embedding for connected compact symmetric spaces with unique isometric embedding.