Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning
This addresses a foundational mathematical flaw in applying Riemannian geometry to robotics tasks with geometric constraints, which is incremental as it clarifies existing methods rather than introducing new ones.
The paper tackles the misuse of Riemannian geometry in robot learning, specifically the 'single tangent space fallacy' where data is projected onto a single Euclidean space, and provides theoretical and experimental evidence of its shortcomings, offering insights for best practices.
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.