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：在高维动态系统的推理与模拟领域，计算仍然面临巨大挑战。要使问题在一般情况下易于处理，需要某种形式的降维。本文提出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降维。具体而言，我们将与预测步骤相关的李雅普诺夫方程投影到低秩矩阵流形上，并利用最近开发的数值稳定的动态低秩积分器进行求解。同时，更新步骤的可处理性源于协方差更新仅变换协方差矩阵的列空间，而该列空间在构造上即为低秩。该算法区别于现有基于集合的方法的关键在于：协方差矩阵的低秩近似是确定性的，而非随机性的。至关重要的是，这使得本方法能够在低秩维度趋近问题真实维度时，复现精确卡尔曼滤波。我们的方法在最坏情况下将计算复杂度从三次（以卡尔曼滤波为基准）降低至状态空间规模的二次方，若状态空间模型满足特定条件，则可实现线性复杂度。通过在经典数据同化和时空回归任务中的系列实验表明，相对于精确卡尔曼滤波，本方法在均值和协方差误差方面始终优于基于集合的方法，且渐近计算复杂度无额外增加。