単一接空間の誤謬を解き明かす：ロボット学習におけるリーマン幾何学適用の分析と明確化
Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning
ロボット学習でリーマン幾何学を用いる際、データを単一の接空間に射影する「単一接空間の誤謬」が広く見られることを指摘し、その理論的問題点と実験的欠点を明らかにして、正しい適用方法を提案した論文。
著者: Noémie Jaquier, Leonel Rozo, Tamim Asfour
分類: cs.RO, cs.LG
原文アブストラクト
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.
