Kalman Filtering with Adversarial Corruptions

TL;DR

This paper develops a robust Kalman filtering method that provides provable guarantees even when a constant fraction of measurements are adversarially corrupted, addressing heavy-tailed and non-stationary noise.

Contribution

It introduces the first robust linear quadratic estimation algorithm with strong guarantees against adversarial corruptions in a Bayesian setting.

Findings

Guarantees robustness against a constant fraction of corrupted measurements

Handles heavy-tailed and non-stationary noise processes

Robustifies Kalman filter to compete with an optimal algorithm knowing corruptions

Abstract

Here we revisit the classic problem of linear quadratic estimation, i.e. estimating the trajectory of a linear dynamical system from noisy measurements. The celebrated Kalman filter gives an optimal estimator when the measurement noise is Gaussian, but is widely known to break down when one deviates from this assumption, e.g. when the noise is heavy-tailed. Many ad hoc heuristics have been employed in practice for dealing with outliers. In a pioneering work, Schick and Mitter gave provable guarantees when the measurement noise is a known infinitesimal perturbation of a Gaussian and raised the important question of whether one can get similar guarantees for large and unknown perturbations. In this work we give a truly robust filter: we give the first strong provable guarantees for linear quadratic estimation when even a constant fraction of measurements have been adversarially corrupted. This framework can model heavy-tailed and even non-stationary noise processes. Our algorithm robustifies the Kalman filter in the sense that it competes with the optimal algorithm that knows the locations of the corruptions. Our work is in a challenging Bayesian setting where the number of measurements scales with the complexity of what we need to estimate. Moreover, in linear dynamical systems past information decays over time. We develop a suite of new techniques to robustly extract information across different time steps and over varying time scales.