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。

探究在 SO (3) 流形上的离散归一化流用于概率旋转建模

Delving into Discrete Normalizing Flows on SO(3) Manifold for  Probabilistic Rotation Modeling

Regressing rotations on SO(3) manifold using deep neural networks is an
important yet unsolved problem. The gap between the Euclidean network output
space and the non-Euclidean SO(3) manifold imposes a severe challenge for
neural network learning in both forward and backward passes. While several
works have proposed different regression-friendly rotation representations,
very few works have been devoted to improving the gradient backpropagating in
the backward pass. In this paper, we propose a manifold-aware gradient that
directly backpropagates into deep network weights. Leveraging Riemannian
optimization to construct a novel projective gradient, our proposed regularized
projective manifold gradient (RPMG) method helps networks achieve new
state-of-the-art performance in a variety of rotation estimation tasks. Our
proposed gradient layer can also be applied to other smooth manifolds such as
the unit sphere. Our project page is at this https URL

本文提出了一种基于流形感知梯度的正则化投影流形梯度（RPMG）方法，旨在解决将 SO（3）流形上的回归问题进行深度学习时所面临挑战。通过运用里曼优化构建新的投影梯度，该方法能够帮助网络在旋转估计任务中取得最新的最高性能。该梯度层也适用于其他平滑流形，如单位球。