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：用于机器人学习的许多工具都是为欧几里得数据设计的。然而，机器人领域的众多应用涉及流形值数据。一个常见例子是姿态；这可以用3×3旋转矩阵或四元数表示，而它们的空间属于非欧几里得流形。在机器人学习中，流形值数据通常通过将流形关联到合适的欧几里得空间来处理，方式包括嵌入流形或将数据投影到一个或多个切空间。这些方法可能导致预测精度低下和算法复杂。本文提出一种直接在流形内部进行回归的"内蕴"方法。该方法在流形上选取合适的概率分布，令其参数成为预测变量（如时间）的函数，然后通过包含核函数的"局部似然"方法对该函数进行非参数估计。我们将此方法命名为核化似然估计。该方案概念简洁，普遍适用于不同流形。我们针对机器人应用中常见的三种不同类型流形值数据实现了该方法。实验结果表明，其预测精度优于基于投影的算法。