Reparameterizing Distributions on Lie Groups
This work addresses a gap in machine learning for handling non-Euclidean spaces like Lie groups, enabling realistic uncertainty estimates in applications such as pose estimation, though it is incremental in extending reparameterization techniques to new domains.
The paper tackles the problem of learning probability distributions on Lie groups, which lack a general reparameterization trick for low-variance gradient estimation in deep learning, by defining a framework to create reparameterizable densities on arbitrary Lie groups and demonstrating its application to complex distributions on SO(3) using normalizing flows for Bayesian pose estimation.
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.