Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations

TL;DR

This paper explores the geometric properties of Wasserstein and information geometries in affine deformation models, highlighting their differences in robustness and efficiency for statistical estimation.

Contribution

It introduces a comparative analysis of Wasserstein and information-geometric estimators within affine deformation models, emphasizing Wasserstein's robustness and specific estimator behaviors.

Findings

Wasserstein geometry separates shape and deformation, offering robustness.

Wasserstein estimator aligns with moment estimators for elliptically symmetric models.

Wasserstein estimator matches maximum-likelihood estimator for Gaussian waveforms.

Abstract

Information geometry and Wasserstein geometry are two main structures introduced in a manifold of probability distributions, and they capture its different characteristics. We study characteristics of Wasserstein geometry in the framework of Li and Zhao (2023) for the affine deformation statistical model, which is a multi-dimensional generalization of the location-scale model. We compare merits and demerits of estimators based on information geometry and Wasserstein geometry. The shape of a probability distribution and its affine deformation are separated in the Wasserstein geometry, showing its robustness against the waveform perturbation in exchange for the loss in Fisher efficiency. We show that the Wasserstein estimator is the moment estimator in the case of the elliptically symmetric affine deformation model. It coincides with the information-geometrical estimator (maximum-likelihood estimator) when the waveform is Gaussian. The role of the Wasserstein efficiency is elucidated in terms of robustness against waveform change.