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Inference and simulation in the context of high-dimensional dynamical systemsremain computationally challenging problems. Some form of dimensionalityreduction is required to make the problem tractable in general. In this paper,we propose a novel approximate Gaussian filtering and smoothing method whichpropagates low-rank approximations of the covariance matrices. This isaccomplished by projecting the Lyapunov equations associated with theprediction step to a manifold of low-rank matrices, which are then solved by arecently developed, numerically stable, dynamical low-rank integrator.Meanwhile, the update steps are made tractable by noting that the covarianceupdate only transforms the column space of the covariance matrix, which islow-rank by construction. The algorithm differentiates itself from existingensemble-based approaches in that the low-rank approximations of the covariancematrices are deterministic, rather than stochastic. Crucially, this enables themethod to reproduce the exact Kalman filter as the low-rank dimensionapproaches the true dimensionality of the problem. Our method reducescomputational 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 setof experiments in classical data-assimilation and spatio-temporal regression,we show that the proposed method consistently outperforms the ensemble-basedmethods in terms of error in the mean and covariance with respect to the exactKalman filter. This comes at no additional cost in terms of asymptoticcomputational complexity.

The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions