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理与模拟中，计算始终是极具挑战性的问题。通常需要某种形式的降维才能使问题具有可处理性。本文提出一种新型近似高斯滤波与平滑方法，该方法传播协方差矩阵的低秩近似。其核心思想是将与预测步骤相关的Lyapunov方程投影到低秩矩阵流形上，然后通过近期开发的数值稳定的动态低秩积分器进行求解。同时，利用更新步骤仅变换协方差矩阵列空间（该空间本身即为低秩结构）的性质，使更新步骤具有可计算性。该算法与现有基于集合方法的关键区别在于：其协方差矩阵的低秩近似是确定性的而非随机的。至关重要的是，当低秩维度趋近于问题的真实维度时，该方法能够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卡尔曼滤波的三次方降低为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。通过经典数据同化与时空回归实验，我们证明：相比基于集合的方法，所提方法在相对于精确卡尔曼滤波的均值与协方差误差方面具有持续的优势，而渐近计算复杂度并未增加。