Marginalizing and Conditioning Gaussians onto Linear Approximations of Smooth Manifolds with Applications in Robotics

TL;DR

This paper introduces closed-form solutions for marginalizing and conditioning Gaussians on linear manifolds and extends these to nonlinear manifolds via linearization, with applications in robotics for improved probabilistic modeling.

Contribution

It provides new analytical expressions for Gaussian marginalization and conditioning on non-axis-aligned manifolds, enhancing nonlinear manifold approximation techniques.

Findings

Improved approximation of projected normal distribution with decreasing nonlinearity

Consistent covariance extraction in Koopman SLAM on real-world data

Reliable covariance estimation in constrained GTSAM in simulations

Abstract

We present closed-form expressions for marginalizing and conditioning Gaussians onto linear manifolds, and demonstrate how to apply these expressions to smooth nonlinear manifolds through linearization. Although marginalization and conditioning onto axis-aligned manifolds are well-established procedures, doing so onto non-axis-aligned manifolds is not as well understood. We demonstrate the utility of our expressions through three applications: 1) approximation of the projected normal distribution, where the quality of our linearized approximation increases as problem nonlinearity decreases; 2) covariance extraction in Koopman SLAM, where our covariances are shown to be consistent on a real-world dataset; and 3) covariance extraction in constrained GTSAM, where our covariances are shown to be consistent in simulation.