Wasserstein Distributionally Robust Kalman Filtering

TL;DR

This paper introduces a distributionally robust Kalman filter that optimally estimates states under model uncertainty by leveraging Wasserstein ambiguity sets, resulting in a tractable convex optimization approach.

Contribution

It develops a novel robust filtering method using Wasserstein ambiguity sets, with a tractable convex reformulation and a specialized Frank-Wolfe algorithm.

Findings

The estimator and least favorable distribution form a Nash equilibrium.

The estimation problem reduces to a convex program despite non-convex ambiguity set.

The proposed filter effectively hedges against model risk.

Abstract

We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filter that hedges against model risk.