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。

计算感知的卡尔曼滤波和平滑

Computation-Aware Kalman Filtering and Smoothing

We study a probabilistic numerical method for the solution of both boundary
and initial value problems that returns a joint Gaussian process posterior over
the solution. Such methods have concrete value in the statistics on Riemannian
manifolds, where non-analytic ordinary differential equations are involved in
virtually all computations. The probabilistic formulation permits marginalising
the uncertainty of the numerical solution such that statistics are less
sensitive to inaccuracies. This leads to new Riemannian algorithms for mean
value computations and principal geodesic analysis. Marginalisation also means
results can be less precise than point estimates, enabling a noticeable
speed-up over the state of the art. Our approach is an argument for a wider
point that uncertainty caused by numerical calculations should be tracked
throughout the pipeline of machine learning algorithms.

研究了一种基于概率的数值方法，用于解决边界和初值问题，并返回解的联合高斯过程后验。提出的方法可用于非解析常微分方程上的流形统计学问题，通过较少精度推算的边际化方法，可以使统计结果不那么敏感。该方法可以用于均值计算和主要测地线分析等新的 Riemann 算法，也可以通过结果不那么精确的方法比点估计更快地完成。该方法认为，在整个机器学习算法流程中应跟踪数值计算引起的不确定性。