# Lognormal Distributions and Geometric Averages of Positive Definite Matrices

arXiv: https://arxiv.org/abs/1407.6383 · HTML: https://mlsanity.com/paper/1407.6383/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2014.

## Built on by (most cited first, 7 of 7)

- [Lie Group Decompositions for Equivariant Neural Networks](https://mlsanity.com/paper/2310.11366/) (2023)
- [Differentially Private Fr'echet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric](https://mlsanity.com/paper/2208.04245/) (2022)
- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2307.12438/) (2023)
- [Robust and Differentially Private Principal Component Analysis](https://mlsanity.com/paper/2507.15232/) (2025)
- [Non-parametric regression for robot learning on manifolds](https://mlsanity.com/paper/2310.19561/) (2023)
- [Enhancing Differentially Private Mechanisms via Empirical Bayes](https://mlsanity.com/paper/2606.21107/) (2026)

## Abstract (arXiv)

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of PD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.