# Riemannian Networks over Full-Rank Correlation Matrices

arXiv: https://arxiv.org/abs/2605.19073 · HTML: https://mlsanity.com/paper/2605.19073/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 55 references that are in the map.
- First posted on arXiv 18 May 2026.

## Builds on (most cited first, 30 of 31)

- [NTU RGB+D 120: A Large-Scale Benchmark for 3D Human Activity Understanding](https://mlsanity.com/paper/1905.04757/) (2019)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Channel-wise Topology Refinement Graph Convolution for Skeleton-Based Action Recognition](https://mlsanity.com/paper/2107.12213/) (2021)
- [First-Person Hand Action Benchmark with RGB-D Videos and 3D Hand Pose Annotations](https://mlsanity.com/paper/1704.02463/) (2017)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [MAtt: A Manifold Attention Network for EEG Decoding](https://mlsanity.com/paper/2210.01986/) (2022)
- [Deep Learning on Lie Groups for Skeleton-based Action Recognition](https://mlsanity.com/paper/1612.05877/) (2016)
- [Training Deep Networks with Structured Layers by Matrix Backpropagation](https://mlsanity.com/paper/1509.07838/) (2015)
- [Computationally Tractable Riemannian Manifolds for Graph Embeddings](https://mlsanity.com/paper/2002.08665/) (2020)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Riemannian Residual Neural Networks](https://mlsanity.com/paper/2310.10013/) (2023)
- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [SPDIM: Source-Free Unsupervised Conditional and Label Shift Adaptation in EEG](https://mlsanity.com/paper/2411.07249/) (2024)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [A New Parametrization of Correlation Matrices](https://mlsanity.com/paper/2012.02395/) (2020)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [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)
- [Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry](https://mlsanity.com/paper/2407.02607/) (2024)
- [SMLNet: A SPD Manifold Learning Network for Infrared and Visible Image Fusion](https://mlsanity.com/paper/2411.10679/) (2024)
- [Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure](https://mlsanity.com/paper/2406.09023/) (2024)

## Abstract (arXiv)

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.