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

TL;DR

This survey articulates how learning a Riemannian metric ${\mathfrak{g}}$ on a data manifold ${\mathcal{M}}$ extends classical distance metric learning by incorporating intrinsic geometry, enabling geodesics, parallel transport, and volume-based probabilistic modeling. It organizes the field around explicit and implicit metric parametrizations, outlines how to compute Riemannian objects like ${\mathbf{d}}_{\mathfrak{g}}$, ${\mathrm{Exp}}_{\mathfrak{g}}$, ${\mathfrak{T}}_{\mathfrak{g}}$, and ${\text{vol}}_{\mathfrak{g}}$, and surveys a broad set of applications from temporal trajectory modeling to causal inference and optimal transport. The paper consolidates methods across classification, regression, and distribution-trajectory objectives, and discusses optimization strategies ranging from closed-form solutions to gradient-based and structured approaches. It emphasizes the potential of geometry-aware representations to improve generalization and enable new capabilities, while highlighting computational bottlenecks, theoretical gaps, and the need for software tooling. Overall, Riemannian metric learning emerges as a powerful, flexible paradigm for representation learning that is poised to shape future research and applications, provided its computational and theoretical challenges are addressed.

Abstract

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.

Figures (10)

• Figure 1: On the left, a classic 1-NN using euclidean distance can separate red and blue points. In the middle, a 1-NN using distance metric learning can separate the points on the anisotropic grid, but not a classic 1-NN. On the right, only a 1-NN using Riemannian metric learning can achieve the separation of spiral data.
• Figure 2: Illustration of tangent spaces, geodesics and the Exponential map. The curve $\gamma_{x,v}$ is the geodesic starting from $x\in \mathcal{M}$ with velocity $v\in T_x\mathcal{M}$.
• Figure 3: Effect of the metric on the shape of the minimizing curve in \ref{['eq:dist_opt']}. The effort required for displacement around a point can be modeled in physics as an energy input, which is defined as the metric norm of the local velocity. When retrieving your lost ball from the sea, following a straight trajectory wastes energy for nothing.
• Figure 4: Illustration of the parallel transport along the curve $\gamma_{x,v_1}$ (already represented in \ref{['fig:preli_scheme_tangent_space']}).
• Figure 5: Illustration of a normal and a global coordinate chart for feature representations.

Theorems & Definitions (20)

• Definition 1: Metric space
• Definition 2: Tangent space
• Example 1
• Definition 3: Riemannian manifold
• Example 2
• Definition 4: Riemannian distance
• Example 3
• Remark 1
• Proposition 1
• Proposition 2: Parallel transport