Variational Kalman Filtering with H∞-Based Correction for Robust Bayesian Learning in High Dimensions

In this paper, we address the problem of convergence of sequential variational inference filter (VIF) through the application of a robust variational objective and H
∞
-norm based correction for a linear Gaussian system. As the dimension of state or parameter space grows, performing the full Kalman update with the dense covariance matrix for a large-scale system requires increased storage and computational complexity, making it impractical. The VIF approach, based on mean-field Gaussian variational inference, reduces this burden through the variational approximation to the covariance usually in the form of a diagonal covariance approximation. The challenge is to retain convergence and correct for biases introduced by the sequential VIF steps. We desire a frame-work that improves feasibility while still maintaining reasonable proximity to the optimal Kalman filter as data is assimilated. To accomplish this goal, a H
∞-
norm based optimization perturbs the VIF covariance matrix to improve robustness. This yields a novel VIF-H
∞
recursion that employs consecutive variational inference and H
∞
based optimization steps. We explore the development of this method and investigate a numerical example to illustrate the effectiveness of the proposed filter.