A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares

TL;DR

This work addresses making Gaussian mixture likelihoods compatible with nonlinear least squares in robotics by introducing a Hessian that correctly accounts for the LogSumExp nonlinearity. The Hessian-Sum-Mixture (HSM) derives a GN-based Hessian for each mixture component and aggregates them with the chain-rule-corrected nonlinearities, ensuring all components contribute consistently. Empirical results in toy, 2D/3D point-set registration, and a real SLAM dataset show improved convergence speed and uncertainty characterization, with robustness against overlapping mixture components. The method preserves compatibility with standard solvers like Ceres and is available as open-source software, enabling practical deployment in robust state estimation tasks.

Abstract

This paper proposes a novel Hessian approximation for Maximum a Posteriori estimation problems in robotics involving Gaussian mixture likelihoods. Previous approaches manipulate the Gaussian mixture likelihood into a form that allows the problem to be represented as a nonlinear least squares (NLS) problem. The resulting Hessian approximation used within NLS solvers from these approaches neglects certain nonlinearities. The proposed Hessian approximation is derived by setting the Hessians of the Gaussian mixture component errors to zero, which is the same starting point as for the Gauss-Newton Hessian approximation for NLS, and using the chain rule to account for additional nonlinearities. The proposed Hessian approximation results in improved convergence speed and uncertainty characterization for simulated experiments,and similar performance to the state of the art on real-world experiments. A method to maintain compatibility with existing solvers, such as ceres, is also presented. Accompanying software and supplementary material can be found at https://github.com/decargroup/hessian_sum_mixtures.

Figures (1)

• Figure 1: A comparison of the Hessian approximation for a Gaussian mixture factor consisting of two components, both centered around zero but with different covariances. For a scalar cost and state variable, the Hessian is also scalar. When substituted into a local optimization method, a better Hessian approximation results in better convergence. In particular, the proposed Hessian (orange) is much closer visually to the exact Hessian (green) than the Max-Mixture (blue) and Max-Sum-Mixture (yellow) approaches.