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.

翻译：许多适用于机器人学习的工具都是为欧几里得数据设计的，但机器人领域的诸多应用涉及流形值数据。一个常见例子是姿态表示——可通过3×3旋转矩阵或四元数实现，而这些量所在的空间正是非欧几里得流形。在机器人学习中，处理流形值数据通常采用映射策略，即通过嵌入流形或将数据投影到一个或多个切空间，将流形与合适的欧几里得空间关联。这类方法可能导致预测精度低下和算法繁琐。本文提出了一种直接作用于流形内部的"内蕴式"回归方法。该方法首先在流形上选取合适的概率分布，令其参数作为预测变量（如时间）的函数，随后通过引入核函数的"局部似然"方法对该函数进行非参数估计，我们将其命名为核化似然估计法。该方案概念简洁且可普适应用于不同流形。针对机器人应用中常见的三类流形值数据进行的实验表明，本方法相比基于投影的算法具有更优的预测精度。