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.

翻译：本文针对机器人领域中涉及高斯混合似然的极大后验估计问题，提出了一种新颖的Hessian近似方法。所提出的Hessian矩阵具有更优的收敛特性。现有方法通过将高斯混合似然转换为可表示为非线性最小二乘问题的形式，然而由于未考虑非线性最小二乘求解器未涵盖的附加非线性因素，导致Hessian近似精度不足。本文通过设定高斯混合分量误差的Hessian矩阵为零（与非线性最小二乘中Gauss-Newton Hessian近似的初始条件相同），并利用链式法则处理附加非线性，推导出全新的Hessian近似公式。该近似方法具有更高精度，显著提升了收敛性能，并通过仿真与真实实验验证。同时给出保持与Ceres等现有求解器兼容性的实现方案。配套软件及补充材料见https://github.com/decargroup/hessian_sum_mixtures。