# A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions

arXiv: https://arxiv.org/abs/2302.08175 · HTML: https://mlsanity.com/paper/2302.08175/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 107 references that are in the map.
- First posted on arXiv 16 Feb 2023.

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

- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means](https://mlsanity.com/paper/1904.04017/) (2019)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [On the Variance of the Fisher Information for Deep Learning](https://mlsanity.com/paper/2107.04205/) (2021)
- [Adversarial Robustness via Fisher-Rao Regularization](https://mlsanity.com/paper/2106.06685/) (2021)
- [Gradient Flows in Filtering and Fisher-Rao Geometry](https://mlsanity.com/paper/1710.00064/) (2017)
- [Hilbert geometry of the Siegel disk: The Siegel-Klein disk model](https://mlsanity.com/paper/2004.08160/) (2020)
- [Clustering in Hilbert simplex geometry](https://mlsanity.com/paper/1704.00454/) (2017)
- [Riemannian optimization for non-centered mixture of scaled Gaussian distributions](https://mlsanity.com/paper/2209.03315/) (2022)
- [A New Robust Regression Method Based on Minimization of Geodesic Distances on a Probabilistic Manifold: Application to Power Laws](https://mlsanity.com/paper/1509.06513/) (2015)
- [Optimal Transport vs. Fisher-Rao distance between Copulas for Clustering Multivariate Time Series](https://mlsanity.com/paper/1604.08634/) (2016)
- [The Fisher-Rao Loss for Learning under Label Noise](https://mlsanity.com/paper/2210.16401/) (2022)

## Built on by (most cited first, 11 of 11)

- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models](https://mlsanity.com/paper/2403.10089/) (2024)
- [Geometric Moment Alignment for Domain Adaptation via Siegel Embeddings](https://mlsanity.com/paper/2510.14666/) (2025)
- [Fisher-Rao distance and pullback SPD cone distances between multivariate normal distributions](https://mlsanity.com/paper/2307.10644/) (2023)
- [Fast proxy centers for Jeffreys centroids: The Jeffreys-Fisher-Rao and the inductive Gauss-Bregman centers](https://mlsanity.com/paper/2410.14326/) (2024)
- [Federated Generalised Variational Inference: A Robust Probabilistic Federated Learning Framework](https://mlsanity.com/paper/2502.00846/) (2025)
- [Robust Estimation in metric spaces: Achieving Exponential Concentration with a Fr'echet Median](https://mlsanity.com/paper/2504.14161/) (2025)
- [Non-parametric regression for robot learning on manifolds](https://mlsanity.com/paper/2310.19561/) (2023)
- [Supervised Quadratic Feature Analysis: Information Geometry Approach for Dimensionality Reduction](https://mlsanity.com/paper/2502.00168/) (2025)
- [Rao Differential Privacy](https://mlsanity.com/paper/2508.17135/) (2025)
- [PRISM: A 3D Probabilistic Neural Representation for Interpretable Shape Modeling](https://mlsanity.com/paper/2602.11467/) (2026)

## Abstract (arXiv)

We present a simple method to approximate Rao's distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating Rao's distances between successive nearby normal distributions on the curves by the square root of Jeffreys divergence, the symmetrized Kullback-Leibler divergence. We consider experimentally the linear interpolation curves in the ordinary, natural and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller's isometric embedding of the Fisher-Rao d-variate normal manifold into the cone of (d+1)× (d+1) symmetric positive-definite matrices [Journal of multivariate analysis 35.2 (1990): 223-242]. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information-geometric properties of the Calvo and Oller's isometric embedding.