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.

翻译：本文针对机器人学中涉及高斯混合似然的最大后验估计问题，提出了一种新颖的Hessian矩阵近似方法。先前的研究方法通过将高斯混合似然转化为特定形式，使得问题能够表示为非线性最小二乘问题。然而，这些方法在非线性最小二乘求解器中使用的Hessian近似忽略了一定的非线性因素。本文提出的Hessian近似方法，通过将高斯混合分量误差的Hessian设为零（这与非线性最小二乘中高斯-牛顿Hessian近似的出发点相同），并运用链式法则来考虑额外的非线性因素，从而推导得出。在仿真实验中，所提出的Hessian近似方法带来了更快的收敛速度和改进的不确定性表征；在真实世界实验中，其性能与现有先进方法相当。本文还提出了一种保持与现有求解器（如ceres）兼容性的方法。相关软件及补充材料可在 https://github.com/decargroup/hessian_sum_mixtures 获取。