Real monodromy action Abstract   The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real parameter spaces are of interest in many applications, real monodromy action is investigated here. A naive extension of monodromy action from the complex numbers to the real numbers is shown to be very restrictive. Therefore, we introduce a real monodromy structure which need not be a group but contains tiered characteristics about the real solutions over the parameter space. An algorithm is provided to compute the real monodromy structure. In addition, this real monodromy structure is applied to an example in kinematics which summarizes all the ways performing loops parameterized by leg lengths can cause a mechanism toxc2xa0changexc2xa0poses.

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.