# Riemannian Local Mechanism for SPD Neural Networks

arXiv: https://arxiv.org/abs/2201.10145 · HTML: https://mlsanity.com/paper/2201.10145/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 53 references that are in the map.
- First posted on arXiv 25 Jan 2022.

## Builds on (most cited first, 20 of 20)

- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Res2Net: A New Multi-scale Backbone Architecture](https://mlsanity.com/paper/1904.01169/) (2019)
- [Convolutional Two-Stream Network Fusion for Video Action Recognition](https://mlsanity.com/paper/1604.06573/) (2016)
- [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)
- [H+O: Unified Egocentric Recognition of 3D Hand-Object Poses and Interactions](https://mlsanity.com/paper/1904.05349/) (2019)
- [Fair DARTS: Eliminating Unfair Advantages in Differentiable Architecture Search](https://mlsanity.com/paper/1911.12126/) (2019)
- [Interpretable 3D Human Action Analysis with Temporal Convolutional Networks](https://mlsanity.com/paper/1704.04516/) (2017)
- [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)
- [Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods](https://mlsanity.com/paper/1605.06182/) (2016)
- [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)
- [Transition Forests: Learning Discriminative Temporal Transitions for Action Recognition and Detection](https://mlsanity.com/paper/1607.02737/) (2016)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [Dilated Convolutional Neural Networks for Sequential Manifold-valued Data](https://mlsanity.com/paper/1910.02206/) (2019)
- [ManifoldNorm: Extending normalizations on Riemannian Manifolds](https://mlsanity.com/paper/2003.13869/) (2020)
- [Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach](https://mlsanity.com/paper/1304.4344/) (2013)
- [Neural Architecture Search of SPD Manifold Networks](https://mlsanity.com/paper/2010.14535/) (2020)
- [Multiple Manifolds Metric Learning with Application to Image Set Classification](https://mlsanity.com/paper/1805.11918/) (2018)

## Built on by (most cited first, 22 of 22)

- [SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space](https://mlsanity.com/paper/2312.08200/) (2023)
- [Adaptive Log-Euclidean Metrics for SPD Matrix Learning](https://mlsanity.com/paper/2303.15477/) (2023)
- [Improving Expressivity of GNNs with Subgraph-specific Factor Embedded Normalization](https://mlsanity.com/paper/2305.19903/) (2023)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [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)
- [Powerformer: A Section-adaptive Transformer for Power Flow Adjustment](https://mlsanity.com/paper/2401.02771/) (2024)
- [Riemannian Self-Attention Mechanism for SPD Networks](https://mlsanity.com/paper/2311.16738/) (2023)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [High-Resolution Underwater Camouflaged Object Detection: GBU-UCOD Dataset and Topology-Aware and Frequency-Decoupled Networks](https://mlsanity.com/paper/2602.03591/) (2026)
- [Riemannian Complex Hermit Positive Definite Convolution Network for Polarimetric SAR Image Classification](https://mlsanity.com/paper/2502.08137/) (2025)
- [Learning Topology-Driven Multi-Subspace Fusion for Grassmannian Deep Network](https://mlsanity.com/paper/2511.08628/) (2025)
- [RepSPD: Enhancing SPD Manifold Representation in EEGs via Dynamic Graphs](https://mlsanity.com/paper/2602.22981/) (2026)
- [K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups](https://mlsanity.com/paper/2603.19601/) (2026)
- [ReManNet: A Riemannian Manifold Network for Monocular 3D Lane Detection](https://mlsanity.com/paper/2603.19776/) (2026)
- [Riemannian Networks over Full-Rank Correlation Matrices](https://mlsanity.com/paper/2605.19073/) (2026)
- [A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding](https://mlsanity.com/paper/2606.06104/) (2026)
- [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)

The Symmetric Positive Definite (SPD) matrices have received wide attention for data representation in many scientific areas. Although there are many different attempts to develop effective deep architectures for data processing on the Riemannian manifold of SPD matrices, very few solutions explicitly mine the local geometrical information in deep SPD feature representations. Given the great success of local mechanisms in Euclidean methods, we argue that it is of utmost importance to ensure the preservation of local geometric information in the SPD networks. We first analyse the convolution operator commonly used for capturing local information in Euclidean deep networks from the perspective of a higher level of abstraction afforded by category theory. Based on this analysis, we define the local information in the SPD manifold and design a multi-scale submanifold block for mining local geometry. Experiments involving multiple visual tasks validate the effectiveness of our approach. The supplement and source code can be found in https://github.com/GitZH-Chen/MSNet.git.