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.

本文提出了一种适用于高维动态系统的近似高斯滤波和平滑方法，通过投影预测步骤中与 Lyapunov 方程相关联的低秩矩阵流形来推进低秩协方差矩阵的逼近，从而将算法的复杂度从 Kalman 滤波的立方降至最坏情况下的二次方、并且在满足一定条件下可达到线性复杂度， experiments 中表明该方法在均值和协方差误差方面的表现优于基于集合的方法，成本不增加。