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

TL;DR

This paper introduces a robust variational Kalman filtering method using Hinf-norm correction to improve convergence and robustness in high-dimensional Bayesian learning, addressing computational challenges of traditional Kalman filters.

Contribution

It proposes a novel VIF-Hinf recursion that combines variational inference with Hinf-norm based optimization for enhanced robustness in high-dimensional systems.

Findings

Improved robustness of the variational filter demonstrated in numerical examples.

Reduced computational complexity compared to full Kalman updates.

Maintains proximity to the optimal Kalman filter during sequential data assimilation.

Abstract

In this paper, we address the problem of convergence of sequential variational inference filter (VIF) through the application of a robust variational objective and Hinf-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 framework that improves feasibility while still maintaining reasonable proximity to the optimal Kalman filter as data is assimilated. To accomplish this goal, a Hinf-norm based optimization perturbs the VIF covariance matrix to improve robustness. This yields a novel VIF- Hinf recursion that employs consecutive variational inference and Hinf based optimization steps. We explore the development of this method and investigate a numerical example to illustrate the effectiveness of the proposed filter.