Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning

TL;DR

The paper addresses the misuse of single tangent spaces in robot learning and demonstrates how this flaw distorts modeling of non-Euclidean data. It compares Euclidean GMMs, Tangent GMMs, and Riemannian GMMs on manifolds like spheres and SPD spaces, showing that mixtures of Riemannian Gaussians with multiple tangent spaces yield the best fit to data and respect manifold geometry. The authors provide theoretical clarifications about tangent spaces, injectivity radii, and the non-isometric nature of exponential maps, backed by experiments in density estimation and dynamical systems learning. The work offers practical guidelines for geometry-aware learning in robotics, advocating coordinate-invariant formulations and the use of tangent-bundle machinery rather than single-space projections.

Abstract

In the realm of robotics, numerous downstream robotics tasks leverage machine learning methods for processing, modeling, or synthesizing data. Often, this data comprises variables that inherently carry geometric constraints, such as the unit-norm condition of quaternions representing rigid-body orientations or the positive definiteness of stiffness and manipulability ellipsoids. Handling such geometric constraints effectively requires the incorporation of tools from differential geometry into the formulation of machine learning methods. In this context, Riemannian manifolds emerge as a powerful mathematical framework to handle such geometric constraints. Nevertheless, their recent adoption in robot learning has been largely characterized by a mathematically-flawed simplification, hereinafter referred to as the "single tangent space fallacy". This approach involves merely projecting the data of interest onto a single tangent (Euclidean) space, over which an off-the-shelf learning algorithm is applied. This paper provides a theoretical elucidation of various misconceptions surrounding this approach and offers experimental evidence of its shortcomings. Finally, it presents valuable insights to promote best practices when employing Riemannian geometry within robot learning applications.

Figures (4)

• Figure 1: Transition function between two charts on the sphere (left), and basic Riemannian operations (right) on the sphere $\mathcal{S}^{2}$. The vector $\bm{u}$ lies on the tangent space of $\bm{x}$ such that $\bm{u} = \text{Log}_{\bm{x}}(\bm{\mathrm{y}})$. $\Gamma_{\bm{\mathrm{x}}\rightarrow\bm{\mathrm{y}}}(\bm{u})$ and $\Gamma_{\bm{\mathrm{x}}\rightarrow\bm{\mathrm{y}}}(\bm{v})$ are parallel-transported vectors.
• Figure 2: Datapoints and GMM of Figs. \ref{['subFig:SphereNaiveGMM']}, \ref{['subFig:SphereNaiveMeanGMM']} in the single tangent spaces at the origin with a top view (left) and Fréchet mean (right).
• Figure 3: Illustration of the fallacies $2$, $3$, and $5$ (in red). Left: the blue and red curves depict the geodesic and Euclidean distances, respectively. Middle: the blue arrows represent two velocities w.r.t two different tangent spaces, while the red arrow depicts a linear velocity on the single tangent space. Right: the black curves display Riemannian trajectories, while the blue and red curves show the corresponding distorted projections on a single tangent space.
• Figure 4: Learning first-order DS on the LASA datasets $\mathsf{S}$(top) and $\mathsf{W}$(bottom) projected on $\mathcal{S}^{2}$. The demonstrations are displayed as white curves and the learned vector field is depicted by arrows (color-coded based on the magnitude). Blue trajectories are reproductions starting at the same initial points as the demonstrations, while black trajectories start from randomly-sampled points in their neighborhood. The vector field of Tangent DS is not displayed in (c) as it is learned on the single tangent space in (d) and does not produce valid velocities.