# Geometric Mean Metric Learning

arXiv: https://arxiv.org/abs/1607.05002 · HTML: https://mlsanity.com/paper/1607.05002/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jul 2016.

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

- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Spectral, Probabilistic, and Deep Metric Learning: Tutorial and Survey](https://mlsanity.com/paper/2201.09267/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [Efficient Hierarchical Bayesian Inference for Spatio-temporal Regression Models in Neuroimaging](https://mlsanity.com/paper/2111.01692/) (2021)
- [Adaptive neighborhood Metric learning](https://mlsanity.com/paper/2201.08314/) (2022)
- [Robust Geometric Metric Learning](https://mlsanity.com/paper/2202.11550/) (2022)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [A Riemannian Approach to Ground Metric Learning for Optimal Transport](https://mlsanity.com/paper/2409.10085/) (2024)
- [Learning Log-Determinant Divergences for Positive Definite Matrices](https://mlsanity.com/paper/2104.06461/) (2021)
- [Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry](https://mlsanity.com/paper/2301.13749/) (2023)
- [Vector Transport Free Riemannian LBFGS for Optimization on Symmetric Positive Definite Matrix Manifolds](https://mlsanity.com/paper/2108.11019/) (2021)
- [On a class of geodesically convex optimization problems solved via Euclidean MM methods](https://mlsanity.com/paper/2206.11426/) (2022)
- [Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2307.12438/) (2023)
- [Random matrix theory improved Fr'echet mean of symmetric positive definite matrices](https://mlsanity.com/paper/2405.06558/) (2024)
- [Learning-Rate-Free Stochastic Optimization over Riemannian Manifolds](https://mlsanity.com/paper/2406.02296/) (2024)
- [Multimodal-Aware Weakly Supervised Metric Learning with Self-weighting Triplet Loss](https://mlsanity.com/paper/2102.02670/) (2021)
- [Hierarchical Relationship Alignment Metric Learning](https://mlsanity.com/paper/2103.15107/) (2021)
- [A Novel Splitting Criterion Inspired by Geometric Mean Metric Learning for Decision Tree](https://mlsanity.com/paper/2204.11011/) (2022)
- [Principal Component Classification](https://mlsanity.com/paper/2210.12746/) (2022)
- [Semantic-Enhanced Relational Metric Learning for Recommender Systems](https://mlsanity.com/paper/2406.10246/) (2024)
- [Kolmogorov Arnold Networks (KANs) for Imbalanced Data -- An Empirical Perspective](https://mlsanity.com/paper/2507.14121/) (2025)
- [A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds](https://mlsanity.com/paper/2608.14220/) (2026)

## Abstract (arXiv)

We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This solution possesses several very attractive properties: (i) an innate geometric appeal through the Riemannian geometry of positive definite matrices; (ii) ease of interpretability; and (iii) computational speed several orders of magnitude faster than the widely used LMNN and ITML methods. Furthermore, on standard benchmark datasets, our closed-form solution consistently attains higher classification accuracy.