# A Review on Riemannian Metric Learning: Closer to You than You Imagine

arXiv: https://arxiv.org/abs/2503.05321 · HTML: https://mlsanity.com/paper/2503.05321/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2025.

## Built on by (most cited first, 13 of 13)

- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry](https://mlsanity.com/paper/2505.08087/) (2025)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds](https://mlsanity.com/paper/2603.28764/) (2026)
- [Geodesic Calculus on Implicitly Defined Latent Manifolds](https://mlsanity.com/paper/2510.09468/) (2025)
- [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026)
- [A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Sufficient Convergence Conditions and Mixing Time Analysis under Gaussian Targets](https://mlsanity.com/paper/2603.18640/) (2026)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [Reciprocal Latent Fields for Precomputed Sound Propagation](https://mlsanity.com/paper/2602.06937/) (2026)
- [Disentanglement Beyond Generative Models with Riemannian ICA](https://mlsanity.com/paper/2605.22531/) (2026)
- [World-Task Factorization for Robot Learning](https://mlsanity.com/paper/2606.02027/) (2026)
- [Task-Induced Riemannian Metrics for Vision Transformer Feature Spaces](https://mlsanity.com/paper/2609.27988/) (2026)

## Abstract (arXiv)

Riemannian metric learning is an emerging field in machine learning, unlocking new ways to encode complex data structures beyond traditional distance metric learning. While classical approaches rely on global distances in Euclidean space, they often fall short in capturing intrinsic data geometry. Enter Riemannian metric learning: a powerful generalization that leverages differential geometry to model the data according to their underlying Riemannian manifold. This approach has demonstrated remarkable success across diverse domains, from causal inference and optimal transport to generative modeling and representation learning. In this review, we bridge the gap between classical metric learning and Riemannian geometry, providing a structured and accessible overview of key methods, applications, and recent advances. We argue that Riemannian metric learning is not merely a technical refinement but a fundamental shift in how we think about data representations. Thus, this review should serve as a valuable resource for researchers and practitioners interested in exploring Riemannian metric learning and convince them that it is closer to them than they might imagine-both in theory and in practice.