# Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices

arXiv: https://arxiv.org/abs/2502.01512 · HTML: https://mlsanity.com/paper/2502.01512/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 72 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Builds on (most cited first, 16 of 16)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [BreizhCrops: A Time Series Dataset for Crop Type Mapping](https://mlsanity.com/paper/1905.11893/) (2019)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [An accelerated first-order method for non-convex optimization on manifolds](https://mlsanity.com/paper/2008.02252/) (2020)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [A Rotated Hyperbolic Wrapped Normal Distribution for Hierarchical Representation Learning](https://mlsanity.com/paper/2205.13371/) (2022)
- [Wrapped Distributions on homogeneous Riemannian manifolds](https://mlsanity.com/paper/2204.09790/) (2022)
- [Exponential-wrapped distributions on symmetric spaces](https://mlsanity.com/paper/2009.01983/) (2020)
- [Wrapped Gaussian Process Functional Regression Model for Batch Data on Riemannian Manifolds](https://mlsanity.com/paper/2409.03181/) (2024)
- [A cheat sheet for probability distributions of orientational data](https://mlsanity.com/paper/2412.08934/) (2024)

## Built on by (most cited first, 5 of 5)

- [Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry](https://mlsanity.com/paper/2505.18193/) (2025)
- [A probabilistic view on Riemannian machine learning models for SPD matrices](https://mlsanity.com/paper/2505.02402/) (2025)
- [LieBN: Batch Normalization over Lie Groups](https://mlsanity.com/paper/2607.08783/) (2026)
- [TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching](https://mlsanity.com/paper/2607.16894/) (2026)
- [Riemannian Flow Models with Reinforcement Learning for Molecular Crystal Structure Prediction](https://mlsanity.com/paper/2609.39773/) (2026)

## Abstract (arXiv)

Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the underlying geometry of such data is pivotal, particularly when extending statistical models, like the pervasive Gaussian distribution. In this work, we tackle those issue by focusing on the manifold of symmetric positive definite (SPD) matrices, a key focus in information geometry. We introduce a non-isotropic wrapped Gaussian by leveraging the exponential map, we derive theoretical properties of this distribution and propose a maximum likelihood framework for parameter estimation. Furthermore, we reinterpret established classifiers on SPD through a probabilistic lens and introduce new classifiers based on the wrapped Gaussian model. Experiments on synthetic and real-world datasets demonstrate the robustness and flexibility of this geometry-aware distribution, underscoring its potential to advance manifold-based data analysis. This work lays the groundwork for extending classical machine learning and statistical methods to more complex and structured data.