# A probabilistic view on Riemannian machine learning models for SPD matrices

arXiv: https://arxiv.org/abs/2505.02402 · HTML: https://mlsanity.com/paper/2505.02402/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 5 May 2025.

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

- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space](https://mlsanity.com/paper/2312.08200/) (2023)
- [Exponential-wrapped distributions on symmetric spaces](https://mlsanity.com/paper/2009.01983/) (2020)
- [Riemann2: Learning Riemannian Submanifolds from Riemannian Data](https://mlsanity.com/paper/2503.05540/) (2025)
- [Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2502.01512/) (2025)
- [Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning](https://mlsanity.com/paper/2411.02726/) (2024)

## Abstract (arXiv)

The goal of this paper is to show how different machine learning tools on the Riemannian manifold Pd of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions defined on Pd. We will show how popular classifiers on Pd can be reinterpreted as Bayes Classifiers using these Gaussian distributions. These distributions will also be used for outlier detection and dimension reduction. By showing that those distributions are pervasive in the tools used on Pd, we allow for other machine learning tools to be extended to Pd.