Many machine learning problems involve regressing variables on a non-Euclidean manifold -- e.g. a discrete probability distribution, or the 6D pose of an object. One way to tackle these problems through gradient-based learning is to use a differentiable function that maps arbitrary inputs of a Euclidean space onto the manifold. In this paper, we establish a set of desirable properties for such mapping, and in particular highlight the importance of pre-images connectivity/convexity. We illustrate these properties with a case study regarding 3D rotations. Through theoretical considerations and methodological experiments on a variety of tasks, we review various differentiable mappings on the 3D rotation space, and conjecture about the importance of their local linearity. We show that a mapping based on Procrustes orthonormalization generally performs best among the mappings considered, but that a rotation vector representation might also be suitable when restricted to small angles.

翻译：许多机器学习问题涉及非欧化方块的递减变量 -- -- 例如离散概率分布,或一个物体的6D构成。通过梯度学习来解决这些问题的一个方法就是使用一种不同的功能,将欧化方块空间的任意输入映射到该方块上。在本文中,我们为这种映射建立一套可取的属性,特别是突出预映像连接/凝固的重要性。我们用关于三维旋转的案例研究来说明这些属性。通过对各种任务的理论考虑和方法实验,我们审查三维旋转空间的各种不同绘图,并推测其本地直线的重要性。我们显示,在所考虑的映射中,基于普罗克鲁斯或异常化的映射通常最能发挥最佳作用,但是,在限于小角度的情况下,旋转矢量代表也可能是合适的。