# Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices

arXiv: https://arxiv.org/abs/1412.4172 · HTML: https://mlsanity.com/paper/1412.4172/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2014.

## Built on by (most cited first, 20 of 20)

- [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)
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- [Deep Efficient Continuous Manifold Learning for Time Series Modeling](https://mlsanity.com/paper/2112.03379/) (2021)
- [R-Mixup: Riemannian Mixup for Biological Networks](https://mlsanity.com/paper/2306.02532/) (2023)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [Geometric Learning with Positively Decomposable Kernels](https://mlsanity.com/paper/2310.13821/) (2023)
- [Collaborative Representation for SPD Matrices with Application to Image-Set Classification](https://mlsanity.com/paper/2201.08962/) (2022)
- [Self-supervised Representation Learning on Electronic Health Records with Graph Kernel Infomax](https://mlsanity.com/paper/2209.00655/) (2022)
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- [Disciplined Geodesically Convex Programming](https://mlsanity.com/paper/2407.05261/) (2024)
- [Gram-MMD: A Texture-Aware Metric for Image Realism Assessment](https://mlsanity.com/paper/2604.03064/) (2026)
- [Deep and Shallow Covariance Feature Quantization for 3D Facial Expression Recognition](https://mlsanity.com/paper/2105.05708/) (2021)
- [Riemannian statistics meets random matrix theory: towards learning from high-dimensional covariance matrices](https://mlsanity.com/paper/2203.00204/) (2022)
- [Efficient Failure Pattern Identification of Predictive Algorithms](https://mlsanity.com/paper/2306.00760/) (2023)
- [Expressivity of congruence-based architectures for DNNs on positive-definite matrices](https://mlsanity.com/paper/2606.02490/) (2026)
- [Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects](https://mlsanity.com/paper/2607.28981/) (2026)
- [On the Abundance of Critical Points of the t-SNE Energy](https://mlsanity.com/paper/2609.04379/) (2026)
- [A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddings](https://mlsanity.com/paper/2609.19083/) (2026)

## Abstract (arXiv)

Symmetric Positive Definite (SPD) matrices have become popular to encode image information. Accounting for the geometry of the Riemannian manifold of SPD matrices has proven key to the success of many algorithms. However, most existing methods only approximate the true shape of the manifold locally by its tangent plane. In this paper, inspired by kernel methods, we propose to map SPD matrices to a high dimensional Hilbert space where Euclidean geometry applies. To encode the geometry of the manifold in the mapping, we introduce a family of provably positive definite kernels on the Riemannian manifold of SPD matrices. These kernels are derived from the Gaussian ker- nel, but exploit different metrics on the manifold. This lets us extend kernel-based algorithms developed for Euclidean spaces, such as SVM and kernel PCA, to the Riemannian manifold of SPD matrices. We demonstrate the benefits of our approach on the problems of pedestrian detection, ob- ject categorization, texture analysis, 2D motion segmentation and Diffusion Tensor Imaging (DTI) segmentation.