# The Fisher-Rao geometry of CES distributions

arXiv: https://arxiv.org/abs/2310.01032 · HTML: https://mlsanity.com/paper/2310.01032/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 101 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 20 of 20)

- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1412.4172/) (2014)
- [A Plug&Play P300 BCI Using Information Geometry](https://mlsanity.com/paper/1409.0107/) (2014)
- [Regularized Tyler's Scatter Estimator: Existence, Uniqueness, and Algorithms](https://mlsanity.com/paper/1407.3079/) (2014)
- [Regularized M-estimators of scatter matrix](https://mlsanity.com/paper/1405.2528/) (2014)
- [A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions](https://mlsanity.com/paper/2302.08175/) (2023)
- [Robust Estimation of Structured Covariance Matrix for Heavy-Tailed Elliptical Distributions](https://mlsanity.com/paper/1506.05215/) (2015)
- [HePPCAT: Probabilistic PCA for Data with Heteroscedastic Noise](https://mlsanity.com/paper/2101.03468/) (2021)
- [Robust low-rank covariance matrix estimation with a general pattern of missing values](https://mlsanity.com/paper/2107.10505/) (2021)
- [Learning Graphical Factor Models with Riemannian Optimization](https://mlsanity.com/paper/2210.11950/) (2022)
- [Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models](https://mlsanity.com/paper/1304.3206/) (2013)
- [Riemannian optimization for non-centered mixture of scaled Gaussian distributions](https://mlsanity.com/paper/2209.03315/) (2022)
- [On the Fisher information matrix for multivariate elliptically contoured distributions](https://mlsanity.com/paper/1306.6415/) (2013)
- [Riemannian geometry for Compound Gaussian distributions: application to recursive change detection](https://mlsanity.com/paper/2005.10087/) (2020)
- [Robust Geometric Metric Learning](https://mlsanity.com/paper/2202.11550/) (2022)
- [Geodesic Convexity and Regularized Scatter Estimators](https://mlsanity.com/paper/1607.05455/) (2016)
- [Simultaneous penalized M-estimation of covariance matrices using geodesically convex optimization](https://mlsanity.com/paper/1608.08126/) (2016)
- [Fisher-Rao distance and pullback SPD cone distances between multivariate normal distributions](https://mlsanity.com/paper/2307.10644/) (2023)

## Built on by (most cited first, 5 of 5)

- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Intrinsic Bayesian Cram'er-Rao Bound with an Application to Covariance Matrix Estimation](https://mlsanity.com/paper/2311.04748/) (2023)
- [Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning](https://mlsanity.com/paper/2411.02726/) (2024)
- [Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds](https://mlsanity.com/paper/2501.11555/) (2025)
- [Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning](https://mlsanity.com/paper/2506.10628/) (2025)

## Abstract (arXiv)

When dealing with a parametric statistical model, a Riemannian manifold can naturally appear by endowing the parameter space with the Fisher information metric. The geometry induced on the parameters by this metric is then referred to as the Fisher-Rao information geometry. Interestingly, this yields a point of view that allows for leveragingmany tools from differential geometry. After a brief introduction about these concepts, we will present some practical uses of these geometric tools in the framework of elliptical distributions. This second part of the exposition is divided into three main axes: Riemannian optimization for covariance matrix estimation, Intrinsic Cram'er-Rao bounds, and classification using Riemannian distances.