Exactly Sparse Gaussian Variational Inference with Application to Derivative-Free Batch Nonlinear State Estimation

We present a Gaussian Variational Inference (GVI) technique that can be applied to large-scale nonlinear batch state estimation problems. The main contribution is to show how to fit the best Gaussian to the posterior efficiently by exploiting factorization of the joint likelihood of the state and data, as is common in practical problems. The proposed Exactly Sparse Gaussian Variational Inference (ESGVI) technique stores the inverse covariance matrix, which is typically very sparse (e.g., block-tridiagonal for classic state estimation). We show that the only blocks of the (dense) covariance matrix that are required during the calculations correspond to the non-zero blocks of the inverse covariance matrix, and further show how to calculate these blocks efficiently in the general GVI problem. ESGVI operates iteratively, and while we can use analytical derivatives at each iteration, Gaussian cubature can be substituted, thereby producing an efficient derivative-free batch formulation. ESGVI simplifies to precisely the Rauch-Tung-Striebel (RTS) smoother in the batch linear estimation case, but goes beyond the 'extended' RTS smoother in the nonlinear case since it finds the best-fit Gaussian, not the Maximum A Posteriori (MAP) point solution. We demonstrate the technique on controlled simulation problems and a batch nonlinear Simultaneous Localization and Mapping (SLAM) problem with an experimental dataset.