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

arXiv: https://arxiv.org/abs/2310.07902 · HTML: https://mlsanity.com/paper/2310.07902/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 11 Oct 2023.

## Builds on (most cited first, 17 of 17)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [On-Manifold Preintegration for Real-Time Visual-Inertial Odometry](https://mlsanity.com/paper/1512.02363/) (2015)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Towards Orientation Learning and Adaptation in Cartesian Space](https://mlsanity.com/paper/1907.03918/) (2019)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Orientation Probabilistic Movement Primitives on Riemannian Manifolds](https://mlsanity.com/paper/2110.15036/) (2021)
- [Wrapped Distributions on homogeneous Riemannian manifolds](https://mlsanity.com/paper/2204.09790/) (2022)
- [Geometric Reinforcement Learning For Robotic Manipulation](https://mlsanity.com/paper/2210.08126/) (2022)
- [Learning Deep Robotic Skills on Riemannian manifolds](https://mlsanity.com/paper/2210.15244/) (2022)
- [K-Tangent Spaces on Riemannian Manifolds for Improved Pedestrian Detection](https://mlsanity.com/paper/1403.1056/) (2014)

## Built on by (most cited first, 6 of 6)

- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Fast and Robust Visuomotor Riemannian Flow Matching Policy](https://mlsanity.com/paper/2412.10855/) (2024)
- [Elastic Motion Policy: An Adaptive Dynamical System for Robust and Efficient One-Shot Imitation Learning](https://mlsanity.com/paper/2503.08029/) (2025)
- [Reinforcement Learning with Lie Group Orientations for Robotics](https://mlsanity.com/paper/2409.11935/) (2024)
- [Non-parametric regression for robot learning on manifolds](https://mlsanity.com/paper/2310.19561/) (2023)
- [Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces](https://mlsanity.com/paper/2506.01635/) (2025)

## Abstract (arXiv)

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.