高效计算多输出高斯过程后验均值，提升大规模时空数据处理速度
Low-rank computation of the posterior mean in Multi-Output Gaussian Processes
- 利用空间时间分离的核函数构造克罗内克积协方差矩阵
- 通过低秩方法求解大型Stein方程，显著降低计算开销
- 适用于大规模图结构数据，适合需要高效推理的场景
高斯过程（GP）是机器学习与计算科学中一种通用工具。本文研究多输出高斯过程（MOGP），提出高效的低秩方法以计算其后验均值。基于低秩时空数据，假设协方差函数在空间和时间上可分离，从而将协方差矩阵分解为各因子协方差矩阵的克罗内克积。引入典型噪声项后，需求解一个大规模Stein方程来计算后验均值。为此，我们提出结合低秩预条件共轭梯度法（LRPCG）与针对Stein方程优化的KPIK（Sylvester方程求解器）的高效算法。我们在真实街道路网图数据上测试该方法，使用图滤波器作为协方差矩阵。此外，还提出一种基于节点度加权的平均协方差矩阵，在特定假设下可加速收敛。
原文摘要 · Abstract (English)
Gaussian processes (GP) are a versatile tool in machine learning and computational science. We here consider the case of multi-output Gaussian processes (MOGP) and present low-rank approaches for efficiently computing the posterior mean of a MOGP. Starting from low-rank spatio-temporal data we consider a structured covariance function, assuming separability across space and time. This separability, in turn, gives a decomposition of the covariance matrix into a Kronecker product of individual covariance matrices. Incorporating the typical noise term to the model then requires the solution of a large-scale Stein equation for computing the posterior mean. For this, we propose efficient low-rank methods based on a combination of a LRPCG method with the Sylvester equation solver KPIK adjusted for solving Stein equations. We test the developed method on real world street network graphs by using graph filters as covariance matrices. Moreover, we propose a degree-weighted average covariance matrix, which can be employed under specific assumptions to achieve more efficient convergence.
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