# Non-parametric regression for robot learning on manifolds

arXiv: https://arxiv.org/abs/2310.19561 · HTML: https://mlsanity.com/paper/2310.19561/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 62 references that are in the map.
- First posted on arXiv 30 Oct 2023.

## Builds on (most cited first, 18 of 18)

- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Kernelized Movement Primitives](https://mlsanity.com/paper/1708.08638/) (2017)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Energy-Based Models for Anomaly Detection: A Manifold Diffusion Recovery Approach](https://mlsanity.com/paper/2310.18677/) (2023)
- [Gaussians on Riemannian Manifolds: Applications for Robot Learning and Adaptive Control](https://mlsanity.com/paper/1909.05946/) (2019)
- [Towards Orientation Learning and Adaptation in Cartesian Space](https://mlsanity.com/paper/1907.03918/) (2019)
- [Merging Position and Orientation Motion Primitives](https://mlsanity.com/paper/2003.11507/) (2020)
- [A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions](https://mlsanity.com/paper/2302.08175/) (2023)
- [Orientation Probabilistic Movement Primitives on Riemannian Manifolds](https://mlsanity.com/paper/2110.15036/) (2021)
- [Bingham Policy Parameterization for 3D Rotations in Reinforcement Learning](https://mlsanity.com/paper/2202.03957/) (2022)
- [Lognormal Distributions and Geometric Averages of Positive Definite Matrices](https://mlsanity.com/paper/1407.6383/) (2014)
- [Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning](https://mlsanity.com/paper/2310.07902/) (2023)
- [A co-design approach for a rehabilitation robot coach for physical rehabilitation based on the error classification of motion errors](https://mlsanity.com/paper/2111.09729/) (2021)
- [Learning Deep Robotic Skills on Riemannian manifolds](https://mlsanity.com/paper/2210.15244/) (2022)
- [Human-to-Robot Manipulability Domain Adaptation with Parallel Transport and Manifold-Aware ICP](https://mlsanity.com/paper/2208.07588/) (2022)
- [Learning and Extrapolation of Robotic Skills using Task-Parameterized Equation Learner Networks](https://mlsanity.com/paper/2212.08531/) (2022)

## Abstract (arXiv)

Many of the tools available for robot learning were designed for Euclidean data. However, many applications in robotics involve manifold-valued data. A common example is orientation; this can be represented as a 3-by-3 rotation matrix or a quaternion, the spaces of which are non-Euclidean manifolds. In robot learning, manifold-valued data are often handled by relating the manifold to a suitable Euclidean space, either by embedding the manifold or by projecting the data onto one or several tangent spaces. These approaches can result in poor predictive accuracy, and convoluted algorithms. In this paper, we propose an "intrinsic" approach to regression that works directly within the manifold. It involves taking a suitable probability distribution on the manifold, letting its parameter be a function of a predictor variable, such as time, then estimating that function non-parametrically via a "local likelihood" method that incorporates a kernel. We name the method kernelised likelihood estimation. The approach is conceptually simple, and generally applicable to different manifolds. We implement it with three different types of manifold-valued data that commonly appear in robotics applications. The results of these experiments show better predictive accuracy than projection-based algorithms.