Fast Kalman filtering and forward-backward smoothing via a low-rank perturbative approach

Kalman filtering-smoothing is a fundamental tool in statistical time series analysis. However, standard implementations of the Kalman filter-smoother require O(d3) time and O(d2) space per timestep, where d is the dimension of the state variable, and are therefore impractical in high-dimensional problems. In this paper we note that if a relatively small number of observations are available per time step, the Kalman equations may be approximated in terms of a low-rank perturbation of the prior state covariance matrix in the absence of any observations. In many cases this approximation may be computed and updated very efficiently (often in just O(k2d) or O(k2d+kd log d) time and space per timestep, where k is the rank of the perturbation and in general k d), using fast methods from numerical linear algebra. We justify our approach and give bounds on the rank of the perturbation as a function of the desired accuracy. For the case of smoothing we also quantify the error of our algorithm due to the low rank approximation and show that it can be made arbitrarily low at the expense of a moderate computational cost. We describe applications involving smoothing of spatiotemporal neuroscience data.