Uncertainty Propagation on Unimodular Lie Groups Using a Gaussian Approximation
Keywords: Uncertainty Propagation, Lie Groups, Diffusion Model
TL;DR: We analyze two types of stochastic differential equations on Lie groups and develop a method to calculate the conditional distribution approximately for diffusion models
Abstract: We discuss the connection between two definitions of stochastic differential equations (SDEs) on unimodular Lie groups and derive the mean and covariance propagation equations in this work. Starting from an SDE defined on Lie groups via Mckean-Gangolli injection, we first convert it to a parametric SDE in exponential coordinates. The coefficient transform method for the conversion is stated for both Ito's and Stratonovich's interpretation of the SDE. Then we derive a mean and covariance fitting formula for probability distributions on Lie groups defined by a concentrated distribution on the exponential coordinate. It is used to derive the mean and covariance propagation equations for the SDE defined by injection, which coincides with the equations derived from a Fokker-Planck equation. The Gaussian distribution constructed from the mean and covariance can be used for calculating the cost function of diffusion models on Lie groups.
Submission Number: 2
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