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.

高维 Gauss-Markov 模型的推断中，我们提出了一种概率数值方法以解决滤波和平滑中存在的时间和内存复杂性难题，并演示了该方法在大规模气候数据集上的可扩展性。