Normalizing flows (NFs) provide a powerful tool to construct an expressive
distribution by a sequence of trackable transformations of a base distribution
and form a probabilistic model of underlying data. Rotation, as an important
quantity in computer vision, graphics, and robotics, can exhibit many
ambiguities when occlusion and symmetry occur and thus demands such
probabilistic models. Though much progress has been made for NFs in Euclidean
space, there are no effective normalizing flows without discontinuity or
many-to-one mapping tailored for SO(3) manifold. Given the unique non-Euclidean
properties of the rotation manifold, adapting the existing NFs to SO(3)
manifold is non-trivial. In this paper, we propose a novel normalizing flow on
SO(3) by combining a Mobius transformation-based coupling layer and a
quaternion affine transformation. With our proposed rotation normalizing flows,
one can not only effectively express arbitrary distributions on SO(3), but also
conditionally build the target distribution given input observations. Extensive
experiments show that our rotation normalizing flows significantly outperform
the baselines on both unconditional and conditional tasks.

本文提出了一种结合 Möbius 变换 - 环耦合层和四元数仿射变换的正交群上的正则化流方法， 可以有效地表示旋转上的任意分布，并且可以在给定输入观测值的条件下建立目标分布，具有显著的条件和无条件任务性能优势。