# Exponential-wrapped distributions on symmetric spaces

arXiv: https://arxiv.org/abs/2009.01983 · HTML: https://mlsanity.com/paper/2009.01983/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2020.

## Built on by (most cited first, 6 of 6)

- [From Chaos Comes Order: Ordering Event Representations for Object Recognition and Detection](https://mlsanity.com/paper/2304.13455/) (2023)
- [Gaussian Differential Privacy on Riemannian Manifolds](https://mlsanity.com/paper/2311.10101/) (2023)
- [Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2502.01512/) (2025)
- [Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2307.12438/) (2023)
- [A probabilistic view on Riemannian machine learning models for SPD matrices](https://mlsanity.com/paper/2505.02402/) (2025)
- [Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants](https://mlsanity.com/paper/2602.00641/) (2026)

## Abstract (arXiv)

In many applications, the curvature of the space supporting the data makes the statistical modelling challenging. In this paper we discuss the construction and use of probability distributions wrapped around manifolds using exponential maps. These distributions have already been used on specific manifolds. We describe their construction in the unifying framework of affine locally symmetric spaces. Affine locally symmetric spaces are a broad class of manifolds containing many manifolds encountered in data sciences. We show that on these spaces, exponential-wrapped distributions enjoy interesting properties for practical use. We provide the generic expression of the Jacobian appearing in these distributions and compute it on two particular examples: Grassmannians and pseudo-hyperboloids. We illustrate the interest of such distributions in a classification experiment on simulated data.