  Inference and simulation in the context of high-dimensional dynamical systems
remain computationally challenging problems. Some form of dimensionality
reduction is required to make the problem tractable in general. In this paper,
we propose a novel approximate Gaussian filtering and smoothing method which
propagates low-rank approximations of the covariance matrices. This is
accomplished by projecting the Lyapunov equations associated with the
prediction step to a manifold of low-rank matrices, which are then solved by a
recently developed, numerically stable, dynamical low-rank integrator.
Meanwhile, the update steps are made tractable by noting that the covariance
update only transforms the column space of the covariance matrix, which is
low-rank by construction. The algorithm differentiates itself from existing
ensemble-based approaches in that the low-rank approximations of the covariance
matrices are deterministic, rather than stochastic. Crucially, this enables the
method to reproduce the exact Kalman filter as the low-rank dimension
approaches the true dimensionality of the problem. Our method reduces
computational complexity from cubic (for the Kalman filter) to \emph{quadratic}
in the state-space size in the worst-case, and can achieve \emph{linear}
complexity if the state-space model satisfies certain criteria. Through a set
of experiments in classical data-assimilation and spatio-temporal regression,
we show that the proposed method consistently outperforms the ensemble-based
methods in terms of error in the mean and covariance with respect to the exact
Kalman filter. This comes at no additional cost in terms of asymptotic
computational complexity.
