# From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices

arXiv: https://arxiv.org/abs/1407.1120 · HTML: https://mlsanity.com/paper/1407.1120/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jul 2014.

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

- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Optimal Randomized Approximations for Matrix based Renyi's Entropy](https://mlsanity.com/paper/2205.07426/) (2022)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators](https://mlsanity.com/paper/2201.08530/) (2022)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Fun with Flags: Robust Principal Directions via Flag Manifolds](https://mlsanity.com/paper/2401.04071/) (2024)
- [Structure-Preserving Transformers for Sequences of SPD Matrices](https://mlsanity.com/paper/2309.07579/) (2023)
- [Grassmannian learning mutual subspace method for image set recognition](https://mlsanity.com/paper/2111.04352/) (2021)
- [Collaborative Representation for SPD Matrices with Application to Image-Set Classification](https://mlsanity.com/paper/2201.08962/) (2022)
- [GrFormer: A Novel Transformer on Grassmann Manifold for Infrared and Visible Image Fusion](https://mlsanity.com/paper/2506.14384/) (2025)
- [Geometric Moment Alignment for Domain Adaptation via Siegel Embeddings](https://mlsanity.com/paper/2510.14666/) (2025)
- [Enhancing Motor Imagery Decoding in Brain Computer Interfaces using Riemann Tangent Space Mapping and Cross Frequency Coupling](https://mlsanity.com/paper/2310.19198/) (2023)
- [Reconciliation of Statistical and Spatial Sparsity For Robust Image and Image-Set Classification](https://mlsanity.com/paper/2106.00256/) (2021)
- [Supervised Quadratic Feature Analysis: Information Geometry Approach for Dimensionality Reduction](https://mlsanity.com/paper/2502.00168/) (2025)
- [Noise-robust Contrastive Learning for Critical Transition Detection in Dynamical Systems](https://mlsanity.com/paper/2512.12523/) (2025)
- [Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain](https://mlsanity.com/paper/2603.02483/) (2026)
- [Intrinsic effective sample size for manifold-valued Markov chain Monte Carlo via kernel discrepancy](https://mlsanity.com/paper/2605.03266/) (2026)
- [1-Lipschitz Neural Networks on Hadamard Manifolds](https://mlsanity.com/paper/2607.19335/) (2026)

## Abstract (arXiv)

Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many 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 an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.