  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.
