Data Assimilation consists in combining one's model knowledge with a stream of data in order to improve the prediction of the system state. Two successful outlets of that approach are given by the Kalman-Bucy filter and its particle-based analog, the Ensemble Kalman filter. While the former describes the exact filtering density evolution in the case of linear and Gaussian dynamics, in practice the latter is often used in real-world applications such as climate or geosciences, as it is computationally tractable. Despite the intrinsic low-rank structure many real-life systems seem to present, using a small number of particles might lead to significant Monte-Carlo error and stochastic fluctuations. We propose a principled model order reduction of the Kalman-Bucy filter (KBF) by way of the Dynamical Low-Rank (DLR) Approximation method, mimicking a time-evolving truncated Karhunen-Loeven approximation of the filtering density. In essence, leveraging the low-rank structure of the filtering density allows to evolve (an approximation of) it in a dynamically evolving subspace, at reduced computational cost. Under certain assumptions, our framework preserves well-known properties of the KBF (including mean and covariance characterisation), and we also establish error bounds between the true and reduced order model. We also propose a DLR extension of the Ensemble Kalman filter, and show a propagation of chaos property to its rank-reduced mean-field limit.
