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.

翻译：在高维动力系统的推断与仿真背景下，计算仍面临严峻挑战。为使问题普遍可解，通常需要采用某种形式的降维方法。本文提出了一种新颖的近似高斯滤波与平滑方法，该方法通过传播协方差矩阵的低秩近似来实现计算优化。具体而言，我们将预测步骤中涉及的Lyapunov方程投影至低秩矩阵流形，并利用近期发展出的数值稳定的动态低秩积分器进行求解。同时，更新步骤的可行性源于协方差更新仅变换协方差矩阵的列空间——该空间通过构造保持低秩特性。本算法与现有基于集合方法的核心区别在于：协方差矩阵的低秩近似是确定性的而非随机性的。关键优势在于，当低秩维度趋近问题真实维度时，本方法可复现标准卡尔曼滤波器的结果。在最坏情况下，本方法将状态空间规模的计算复杂度从立方级（标准卡尔曼滤波器）降至二次方级；若状态空间模型满足特定条件，更可实现线性复杂度。通过在经典数据同化及时空回归任务中的系列实验表明，相较于基于集合的方法，本方法在均值和协方差误差（相对于精确卡尔曼滤波器）方面持续表现更优，且未增加渐进计算复杂度。