Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. Since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.

翻译：卡尔曼滤波与平滑是对高斯-马尔可夫模型进行高效推理的基础机制。然而，其时间和内存复杂度随状态空间规模呈指数级增长，这在时空回归问题中尤为突出——此类问题的状态维度随空间观测数量增加而扩展。现有近似框架通过协方差矩阵的低秩近似来缓解该问题，但由于未对计算近似引入的误差进行建模，其预测不确定性估计可能过于乐观。本文提出一种用于高维高斯-马尔可夫模型推理的概率数值方法，旨在缓解上述扩展性问题。我们的无矩阵迭代算法利用GPU加速，并关键性地实现了计算成本与预测不确定性之间的可调节权衡。最后，我们在大规模气候数据集上验证了该方法可扩展性。