# Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

arXiv: https://arxiv.org/abs/1605.06182 · HTML: https://mlsanity.com/paper/1605.06182/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2016.

## Built on by (most cited first, 29 of 29)

- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals](https://mlsanity.com/paper/2303.05798/) (2023)
- [Wasserstein Distance Rivals Kullback-Leibler Divergence for Knowledge Distillation](https://mlsanity.com/paper/2412.08139/) (2024)
- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Adaptive Log-Euclidean Metrics for SPD Matrix Learning](https://mlsanity.com/paper/2303.15477/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Temporal-attentive Covariance Pooling Networks for Video Recognition](https://mlsanity.com/paper/2110.14381/) (2021)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Discriminative Noise Robust Sparse Orthogonal Label Regression-based Domain Adaptation](https://mlsanity.com/paper/2101.04563/) (2021)
- [Riemannian Self-Attention Mechanism for SPD Networks](https://mlsanity.com/paper/2311.16738/) (2023)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Learning Log-Determinant Divergences for Positive Definite Matrices](https://mlsanity.com/paper/2104.06461/) (2021)
- [Collaborative Representation for SPD Matrices with Application to Image-Set Classification](https://mlsanity.com/paper/2201.08962/) (2022)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
- [Nested subspace learning with flags](https://mlsanity.com/paper/2502.06022/) (2025)
- [GrFormer: A Novel Transformer on Grassmann Manifold for Infrared and Visible Image Fusion](https://mlsanity.com/paper/2506.14384/) (2025)
- [DreamNet: A Deep Riemannian Network based on SPD Manifold Learning for Visual Classification](https://mlsanity.com/paper/2206.07967/) (2022)
- [MPEC: Manifold-Preserved EEG Classification via an Ensemble of Clustering-Based Classifiers](https://mlsanity.com/paper/2504.21427/) (2025)
- [Learning Chebyshev Basis in Graph Convolutional Networks for Skeleton-based Action Recognition](https://mlsanity.com/paper/2104.05482/) (2021)
- [Discriminative Supervised Subspace Learning for Cross-modal Retrieval](https://mlsanity.com/paper/2201.11843/) (2022)
- [Local Spherical Harmonics Improve Skeleton-Based Hand Action Recognition](https://mlsanity.com/paper/2308.10557/) (2023)
- [LGL-BCI: A Motor-Imagery-Based Brain-Computer Interface with Geometric Learning](https://mlsanity.com/paper/2310.08051/) (2023)
- [DCSCR: A Class-Specific Collaborative Representation based Network for Image Set Classification](https://mlsanity.com/paper/2508.12745/) (2025)
- [Subspace Kernel Learning on Tensor Sequences](https://mlsanity.com/paper/2603.19546/) (2026)
- [Expressivity of congruence-based architectures for DNNs on positive-definite matrices](https://mlsanity.com/paper/2606.02490/) (2026)
- [Towards Robust EEG Decoding Based on Riemannian Self-Attention](https://mlsanity.com/paper/2606.25456/) (2026)

## Abstract (arXiv)

Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.