Diffusion Means and Heat Kernel on Manifolds

TL;DR

This paper introduces diffusion means as a new way to define location statistics on manifold data spaces using heat kernel expansion, enabling calculations on various spaces and addressing issues like smeariness in directional data.

Contribution

It defines diffusion means on manifolds via isotropic diffusion, providing a practical method for their computation using heat kernel expansion on known spaces.

Findings

Diffusion means can be computed on multiple manifolds using heat kernel expansion.

Application to directional data shows the sample Fréchet mean exhibits smeariness.

Method extends the toolkit for statistical analysis on manifold data.

Abstract

We introduce diffusion means as location statistics on manifold data spaces. A diffusion mean is defined as the starting point of an isotropic diffusion with a given diffusivity. They can therefore be defined on all spaces on which a Brownian motion can be defined and numerical calculation of sample diffusion means is possible on a variety of spaces using the heat kernel expansion. We present several classes of spaces, for which the heat kernel is known and sample diffusion means can therefore be calculated. As an example, we investigate a classic data set from directional statistics, for which the sample Fr\'echet mean exhibits finite sample smeariness.