Non-parametric regression for robot learning on manifolds

TL;DR

This paper introduces an intrinsic non-parametric regression method for manifold-valued data in robotics, directly modeling data on the manifold to improve predictive accuracy over traditional Euclidean-based approaches.

Contribution

It proposes a kernelised likelihood estimation method that operates directly on manifolds, providing a simple, general, and more accurate alternative to existing projection-based techniques.

Findings

Outperforms projection-based algorithms in predictive accuracy

Applicable to various common manifolds in robotics

Demonstrated with three types of manifold-valued data

Abstract

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.