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.

翻译：在机器人领域，众多下游机器人任务依赖机器学习方法进行数据处理、建模或合成。这些数据往往包含固有几何约束的变量，例如表示刚体方向四元数的单位范数条件，或刚度与可操作椭球的正定性。有效处理此类几何约束，需要将微分几何工具融入机器学习方法的表述中。在此背景下，黎曼流形成为应对这些几何约束的强有力数学框架。然而，该方法近期在机器人学习中的应用，普遍存在一种数学上有缺陷的简化，本文称之为“单一切空间谬误”。该方法仅将目标数据投影至单个切（欧几里得）空间，在此空间上应用现成的学习算法。本文从理论上阐释了这一方法相关的各种误解，并通过实验证据揭示其缺陷。最后，本文提出宝贵见解，以促进在机器人学习应用中采用黎曼几何的最佳实践。