Reparameterizable densities are an important way to learn probability distributions in a deep learning setting. For many distributions it is possible to create low-variance gradient estimators by utilizing a `reparameterization trick'. Due to the absence of a general reparameterization trick, much research has recently been devoted to extend the number of reparameterizable distributional families. Unfortunately, this research has primarily focused on distributions defined in Euclidean space, ruling out the usage of one of the most influential class of spaces with non-trivial topologies: Lie groups. In this work we define a general framework to create reparameterizable densities on arbitrary Lie groups, and provide a detailed practitioners guide to further the ease of usage. We demonstrate how to create complex and multimodal distributions on the well known oriented group of 3D rotations, $\operatorname{SO}(3)$, using normalizing flows. Our experiments on applying such distributions in a Bayesian setting for pose estimation on objects with discrete and continuous symmetries, showcase their necessity in achieving realistic uncertainty estimates.

本文提出了一种通用的框架，以在任意 Lie 群上创建可重参数化密度，并提供了详细的从业者指南，展示了如何在面向3D旋转的定向群$SO(3)$上使用归一化流创建复杂的和多模态的分布，其应用于具有离散和连续对称性的对象的姿态估计中展示了它们在实现现实情境下的不确定性估计方面的必要性。

李群上的重参数化分布