Delving into Discrete Normalizing Flows on SO Manifold for Probabilistic Rotation Modeling

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.