# Riemannian optimization for non-centered mixture of scaled Gaussian distributions

arXiv: https://arxiv.org/abs/2209.03315 · HTML: https://mlsanity.com/paper/2209.03315/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 53 references that are in the map.
- First posted on arXiv 7 Sep 2022.

## Builds on (most cited first, 7 of 7)

- [SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python](https://mlsanity.com/paper/1907.10121/) (2019)
- [BreizhCrops: A Time Series Dataset for Crop Type Mapping](https://mlsanity.com/paper/1905.11893/) (2019)
- [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)
- [Robust Estimation of Structured Covariance Matrix for Heavy-Tailed Elliptical Distributions](https://mlsanity.com/paper/1506.05215/) (2015)
- [Shrinking the eigenvalues of M-estimators of covariance matrix](https://mlsanity.com/paper/2006.10005/) (2020)
- [Riemannian geometry for Compound Gaussian distributions: application to recursive change detection](https://mlsanity.com/paper/2005.10087/) (2020)

## Built on by (most cited first, 4 of 4)

- [A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions](https://mlsanity.com/paper/2302.08175/) (2023)
- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [Classification of Buried Objects from Ground Penetrating Radar Images by using Second Order Deep Learning Models](https://mlsanity.com/paper/2410.07117/) (2024)
- [Weakly supervised covariance matrices alignment through Stiefel matrices estimation for MEG applications](https://mlsanity.com/paper/2402.03345/) (2024)

## Abstract (arXiv)

This paper studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated to this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first one is the minimization of a regularized negative log-likelihood (NLL). The latter makes the trade-off between a white Gaussian distribution and the NC-MSG. Conditions on the regularization are given so that the existence of a minimum to this problem is guaranteed without assumptions on the samples. Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This divergence enables us to define a minimization problem to compute centers of mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is leveraged to solve this second minimization problem. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a Nearest centroid classifier is implemented leveraging the KL divergence and its associated center of mass. Applied on the large scale dataset Breizhcrops, this classifier shows good accuracies as well as robustness to rigid transformations of the test set.