# Consistent Estimation of a Class of Distances Between Covariance Matrices

arXiv: https://arxiv.org/abs/2409.11761 · HTML: https://mlsanity.com/paper/2409.11761/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 8 of 8)

- [Toeplitz Inverse Covariance-Based Clustering of Multivariate Time Series Data](https://mlsanity.com/paper/1706.03161/) (2017)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)
- [Random Matrix Improved Covariance Estimation for a Large Class of Metrics](https://mlsanity.com/paper/1902.02554/) (2019)
- [Random matrix-improved estimation of covariance matrix distances](https://mlsanity.com/paper/1810.04534/) (2018)
- [Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered Gaussian Distributions](https://mlsanity.com/paper/1903.03447/) (2019)
- [On the Resolution Probability of Conditional and Unconditional Maximum Likelihood DoA Estimation](https://mlsanity.com/paper/2008.00726/) (2020)
- [Probability of Resolution of MUSIC and g-MUSIC: An Asymptotic Approach](https://mlsanity.com/paper/2106.08738/) (2021)
- [Asymptotics of Distances Between Sample Covariance Matrices](https://mlsanity.com/paper/2312.14584/) (2023)

## Abstract (arXiv)

This work considers the problem of estimating the distance between two covariance matrices directly from the data. Particularly, we are interested in the family of distances that can be expressed as sums of traces of functions that are separately applied to each covariance matrix. This family of distances is particularly useful as it takes into consideration the fact that covariance matrices lie in the Riemannian manifold of positive definite matrices, thereby including a variety of commonly used metrics, such as the Euclidean distance, Jeffreys' divergence, and the log-Euclidean distance. Moreover, a statistical analysis of the asymptotic behavior of this class of distance estimators has also been conducted. Specifically, we present a central limit theorem that establishes the asymptotic Gaussianity of these estimators and provides closed form expressions for the corresponding means and variances. Empirical evaluations demonstrate the superiority of our proposed consistent estimator over conventional plug-in estimators in multivariate analytical contexts. Additionally, the central limit theorem derived in this study provides a robust statistical framework to assess of accuracy of these estimators.