# Riemannian Metric Learning for Symmetric Positive Definite Matrices

arXiv: https://arxiv.org/abs/1501.02393 · HTML: https://mlsanity.com/paper/1501.02393/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2015.

## Built on by (most cited first, 8 of 8)

- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023)
- [Multi-manifold Attention for Vision Transformers](https://mlsanity.com/paper/2207.08569/) (2022)
- [Learning residue level protein dynamics with multiscale Gaussians](https://mlsanity.com/paper/2509.01038/) (2025)
- [Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2102.00667/) (2021)
- [A Spectral Framework for Evaluating Geodesic Distances Between Graphs](https://mlsanity.com/paper/2406.10500/) (2024)
- [A Pseudo Global Fusion Paradigm-Based Cross-View Network for LiDAR-Based Place Recognition](https://mlsanity.com/paper/2508.08917/) (2025)
- [On the Importance of Behavioral Nuances: Amplifying Non-Obvious Motor Noise Under True Empirical Considerations May Lead to Briefer Assays and Faster Classification Processes](https://mlsanity.com/paper/2508.12742/) (2025)

## Abstract (arXiv)

Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced by Riemannian geometry. In this work, we focus on the log-Euclidean Riemannian geometry and propose a data-driven approach for learning Riemannian metrics/geodesic distances for SPD matrices. We show that the geodesic distance learned using the proposed approach performs better than various existing distance measures when evaluated on face matching and clustering tasks.