揭示单一切空间谬误：在机器人学习中应用黎曼几何的分析与澄清
Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning
AI总结：
本文揭示了机器人学习中将流形数据简单投影到单一切空间的“单一切空间谬误”，通过理论分析与实验证据阐明其缺陷，并提出了应用黎曼几何的最佳实践。
摘要：
在机器人学领域，众多下游机器人任务利用机器学习方法进行数据的处理、建模或合成。这些数据通常包含固有的几何约束变量，例如表示刚体姿态的四元数的单位范数条件，或刚度与可操作性椭球的正定性。为了有效处理此类几何约束，需要将微分几何工具引入机器学习方法的构建中。在此背景下，黎曼流形成为处理此类几何约束的强大数学框架。然而，近期其在机器人学习中的应用主要表现为一种数学上有缺陷的简化，下文称为“单一切空间谬误”。该方法仅将感兴趣的数据投影到单一的切（欧几里得）空间上，并在该空间上应用现成的学习算法。本文对围绕该方法的诸多误解进行了理论阐释，并提供了其缺陷的实验证据。最后，本文提出了有价值的见解，以促进在机器人学习应用中采用黎曼几何时的最佳实践。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
