# Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations

arXiv: https://arxiv.org/abs/2307.12508 · HTML: https://mlsanity.com/paper/2307.12508/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 58 references that are in the map.
- First posted on arXiv 24 Jul 2023.

## Builds on (most cited first, 13 of 13)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Information Newton's flow: second-order optimization method in probability space](https://mlsanity.com/paper/2001.04341/) (2020)
- [A fast approach to optimal transport: The back-and-forth method](https://mlsanity.com/paper/1905.12154/) (2019)
- [Optimal transport natural gradient for statistical manifolds with continuous sample space](https://mlsanity.com/paper/1805.08380/) (2018)
- [Regularized M-estimators of scatter matrix](https://mlsanity.com/paper/1405.2528/) (2014)
- [When Optimal Transport Meets Information Geometry](https://mlsanity.com/paper/2206.14791/) (2022)
- [Information Geometry Connecting Wasserstein Distance and Kullback-Leibler Divergence via the Entropy-Relaxed Transportation Problem](https://mlsanity.com/paper/1709.10219/) (2017)
- [Wasserstein Training of Boltzmann Machines](https://mlsanity.com/paper/1507.01972/) (2015)

## Built on by (most cited first, 3 of 3)

- [Wasserstein-Cram'er-Rao Theory of Unbiased Estimation](https://mlsanity.com/paper/2511.07414/) (2025)
- [On the attainment of the Wasserstein--Cramer--Rao lower bound](https://mlsanity.com/paper/2506.12732/) (2025)
- [A Transport-Based Geometry of Belief-Cost](https://mlsanity.com/paper/2606.21585/) (2026)

## Abstract (arXiv)

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.