Riemannian Metric Learning: Closer to You than You Imagine (2503.05321v1)

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.

• The paper reviews Riemannian metric learning, explaining how it extends classical metric learning by using differential geometry to capture complex data structures.
• Riemannian metric learning incorporates concepts like manifolds, metrics, and geodesics, often implemented using tools like the Python package geomstats.
• This approach has diverse applications in fields such as image processing, robot learning, generative modeling, and domain adaptation due to its ability to model local data geometry.

Riemannian metric learning is a generalization of classical metric learning techniques that leverages the principles of differential geometry to model data according to its underlying Riemannian manifold. The paper "Riemannian Metric Learning: Closer to You than You Imagine" (2503.05321) offers a review of this emerging field, bridging classical methods with advanced geometric techniques applicable in areas such as causal inference, optimal transport, generative modeling, and representation learning.

Classical vs. Riemannian Metric Learning

Classical metric learning typically seeks a fixed Mahalanobis metric by learning a transformation optimized for data separation. However, this approach is limited in capturing complex, high-dimensional datasets due to its inability to adapt locally to the data's geometric structure. Riemannian metric learning addresses these limitations by allowing the metric to vary across the manifold, thus capturing local geometries and curvatures. This paradigm shift incorporates tools like geodesics, exponential maps, parallel transport, and curvature into learning tasks.

Core Riemannian Concepts and Tools

Riemannian metric learning uses several key concepts and mathematical tools rooted in differential geometry:

• Riemannian Manifold: The input space is treated as a Riemannian manifold $M$, a smooth geometric space where each point has a tangent space.
• Riemannian Metric: A smoothly varying inner product $g$ (a positive definite matrix) is defined on each tangent space, facilitating local measurements of lengths and angles, represented as a matrix field $g: M \rightarrow \mathbb{R}^{d \times d}$.
• Geodesics: These are generalizations of straight lines on a manifold, locally minimizing distance and defined by the geodesic equation $\nabla_{\gamma'} \gamma' = 0$.
• Tangent Space ($T_pM$): A vector space that locally approximates the manifold at a point $p$.
• Exponential Map ($\text{Exp}_p(v)$): Maps a tangent vector $v$ at a point $p$ to a point on the manifold along the geodesic starting at $p$ with initial velocity $v$.
• Logarithm Map ($\text{Log}_p(x)$): The inverse of the exponential map, it maps a point $x$ on the manifold to a tangent vector at $p$, whose geodesic connects $p$ to $x$.
• Parallel Transport: A method to move vectors along the manifold while preserving length and angle, crucial for transfer learning across complex data structures.
• Covariant Derivative ($\nabla$): A generalization of the derivative for vector fields on manifolds.
• Metric Volume: Derived from the determinant of the Riemannian metric, it measures volume on the manifold and is used in integration and defining probability distributions.
• Ricci Curvature: A measure of the intrinsic curvature of a Riemannian manifold.

Geomstats: A Python Package for Implementation

The geomstats package provides a Python implementation for computations on Riemannian manifolds (1805.08308). It supports various manifolds like hyperspheres, hyperbolic spaces, SPD matrices, and Lie groups, equipped with Riemannian metrics, exponential and logarithm maps, and geodesics. The package facilitates geometric statistics, including Fréchet mean computation and tangent PCA. It also supports the integration of Riemannian geometry in deep learning, offering loss functions and modified Keras/TensorFlow versions for models constrained on manifolds. Example applications include neural network weight constraints, visualization on hyperbolic space, brain connectome classification, robot arm trajectory interpolation, and pose estimation with CNNs.

Applications and Related Work

Riemannian metric learning has found applications in diverse fields:

• Image and Video Processing: Utilized in face recognition from video by learning discriminant Riemannian metrics (1608.04200). SPD matrices, common in computer vision, benefit from data-driven Riemannian metrics for tasks like face matching and clustering (1501.02393).
• Domain Adaptation: Riemannian metrics are used to align second-order statistics (covariances) in domain adaptation problems, accounting for the non-positive curvature of Riemannian manifolds (1705.08180).
• Robot Learning and Adaptive Control: Integrates Riemannian geometry and probabilistic representations, applying Gaussian distributions on Riemannian manifolds for tasks like prosthetic hand control and underwater robot teleoperation (1909.05946).
• Generative Modeling: Optimal transport-based models learn metric tensors on Riemannian manifolds, improving trajectory inference in areas like scRNA data analysis and bird migration patterns (2205.09244).

Challenges and Future Directions

Despite its advantages, Riemannian metric learning faces computational challenges due to the complexity of manifold geometry and non-convex optimization. Efficient algorithms and theoretical frameworks are needed to streamline these processes. Enhanced software tools are also essential for broadening accessibility to practitioners across various fields.

In summary, Riemannian metric learning extends classical metric learning by incorporating differential geometry, enabling more nuanced data modeling. Key concepts include Riemannian manifolds, metrics, geodesics, and tangent spaces. Python packages like geomstats facilitate implementation. The field is applied in diverse areas such as image processing, robotics, and generative modeling, with ongoing research focused on addressing computational complexities and enhancing accessibility.