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

In the realm of robotics, numerous downstream robotics tasks leverage machine learning methods for process- ing, 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 short- comings. Finally, it presents valuable insights to promote best practices when employing Riemannian geometry within robot learning applications. This work was supported by the Carl Zeiss Foundation under the project JuBot and the European Union’s Horizon Europe Framework Programme under grant agreement No 101070596 (euROBIN). *These authors contributed equally (listed in alphabetical order). 1Institute for Anthropomatics and Robotics, Karlsruhe Institute of Tech- nology, Karlsruhe, Germany. noemie.jaquier@kit.edu 2Bosch Center for Artificial Intelligence. Renningen, Germany.