Diffusion Means in Geometric Spaces

TL;DR

This paper introduces the diffusion mean, a new statistical measure for distributions on non-linear geometric spaces, extending the Fréchet mean by incorporating a diffusion process parameter and analyzing its properties.

Contribution

It proposes the diffusion mean as a novel extension of the Fréchet mean, including theoretical properties, asymptotic behavior, and applications to spherical data.

Findings

Diffusion mean generalizes the Fréchet mean with a diffusion parameter.

The diffusion estimator is shown to be strongly consistent under certain conditions.

Experimental results indicate improved convergence rates over the Fréchet mean.

Abstract

We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fr\'echet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter $t$, which admits the interpretation of the allowed variance of the diffusion. The diffusion $t$-mean of a distribution $X$ is the most likely origin of a Brownian motion at time $t$, given the end-point distribution $X$. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere $\mathbb S^n$. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fr\'echet mean. Here, we additionally estimate $t$ and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.