# A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices

arXiv: https://arxiv.org/abs/1805.11204 · HTML: https://mlsanity.com/paper/1805.11204/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2018.

## Built on by (most cited first, 18 of 18)

- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
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- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
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- [Adaptive Log-Euclidean Metrics for SPD Matrix Learning](https://mlsanity.com/paper/2303.15477/) (2023)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [Deep Efficient Continuous Manifold Learning for Time Series Modeling](https://mlsanity.com/paper/2112.03379/) (2021)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [A Lie Group Approach to Riemannian Batch Normalization](https://mlsanity.com/paper/2403.11261/) (2024)
- [RMLR: Extending Multinomial Logistic Regression into General Geometries](https://mlsanity.com/paper/2409.19433/) (2024)
- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)
- [Deep Geometric Learning with Monotonicity Constraints for Alzheimer's Disease Progression](https://mlsanity.com/paper/2310.03353/) (2023)
- [Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry](https://mlsanity.com/paper/2407.02607/) (2024)
- [GeoDynamics: A Geometric State-Space Neural Network for Understanding Brain Dynamics on Riemannian Manifolds](https://mlsanity.com/paper/2601.13570/) (2026)
- [Riemannian Complex Hermit Positive Definite Convolution Network for Polarimetric SAR Image Classification](https://mlsanity.com/paper/2502.08137/) (2025)
- [Towards Robust EEG Decoding Based on Riemannian Self-Attention](https://mlsanity.com/paper/2606.25456/) (2026)
- [LieBN: Batch Normalization over Lie Groups](https://mlsanity.com/paper/2607.08783/) (2026)
- [Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold](https://mlsanity.com/paper/2609.30487/) (2026)

## Abstract (arXiv)

In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CNNs) can be extended to learn with graph-based data. In this work, we study the setting where the data (or measurements) are ordered, longitudinal or temporal in nature and live on a Riemannian manifold -- this setting is common in a variety of problems in statistical machine learning, vision and medical imaging. We show how recurrent statistical recurrent network models can be defined in such spaces. We give an efficient algorithm and conduct a rigorous analysis of its statistical properties. We perform extensive numerical experiments demonstrating competitive performance with state of the art methods but with significantly less number of parameters. We also show applications to a statistical analysis task in brain imaging, a regime where deep neural network models have only been utilized in limited ways.