  Many problems in machine learning involve regressing outputs that do not lie
on a Euclidean space, such as a discrete probability distribution, or the pose
of an object. An approach to tackle these problems through gradient-based
learning consists in including in the deep learning architecture a
differentiable function mapping arbitrary inputs of a Euclidean space onto this
manifold. In this work, we establish a set of properties that such mapping
should satisfy to allow proper training, and illustrate it in the case of 3D
rotations. Through theoretical considerations and methodological experiments on
a variety of tasks, we compare various differentiable mappings on the 3D
rotation space, and conjecture about the importance of the local linearity of
the mapping. We notably show that a mapping based on Procrustes
orthonormalization of a 3x3 matrix generally performs best among the ones
considered, but that rotation-vector representation might also be suitable when
restricted to small angles.
