Reparameterizing Distributions on Lie Groups Reparameterizable densities are an important way to learn probability distributions in a deep learning setting. For many distributions it is possible to create lowvariance gradient estimators by utilizing a reparameterization tricku0027. 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 nontrivial 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, SO3, 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.

Nonparametric MixedManifold Products using Multiscale Kernel Densities We extend the core operation of nonparametric belief propagation (NBP), also known as multiscale sequential Gibbs sampling, to approximate products of kernel density estimated beliefs that reside on some manifold. The original algorithm, though multidimensional, implicitly assumes the beliefs to reside on the Euclidean mathbbRd space only. The proposed extension generalizes to any mixture of Riemannian manifolds, provided the primary operationsxe2x80x94addition and subtractionxe2x80x94 are defined. Our motivation is primarily focused on stateestimation using nonGaussian factor graphs for multimodal simultaneous localization and mapping in robotics. The paper presents the method as well as simulation and experimental results for validation. Our implementation is publicly available and allows for expansion with userdefined manifold mixtures.