Reparameterizing Distributions on Lie Groups

TL;DR

This paper introduces a general framework for creating reparameterizable probability densities on Lie groups, enabling advanced Bayesian modeling of rotations and symmetries with improved uncertainty estimation.

Contribution

It develops a novel method to define reparameterizable densities on Lie groups, extending the reparameterization trick beyond Euclidean spaces, with practical applications in pose estimation.

Findings

Effective reparameterization on SO(3) using normalizing flows

Improved uncertainty estimates in Bayesian pose estimation

Demonstrated applicability to complex, multimodal distributions

Abstract

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.