A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares

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.