  This paper proposes a novel Hessian approximation for Maximum a Posteriori
estimation problems in robotics involving Gaussian mixture likelihoods. The
proposed Hessian leads to better convergence properties. Previous approaches
manipulate the Gaussian mixture likelihood into a form that allows the problem
to be represented as a nonlinear least squares (NLS) problem. However, they
result in an inaccurate Hessian approximation due to additional nonlinearities
that are not accounted for in NLS solvers. 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 is more accurate, resulting
in improved convergence properties that are demonstrated on simulated and
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.
