Gaussian Process (GP) models provide a flexible framework for prediction and uncertainty quantification. For most covariance functions, however, exact GP prediction with $n$ points scales as $\mathcal{O}(n^3)$, making it prohibitively expensive for large datasets or large numbers of prediction points. While nearest neighbor-based prediction can work well in certain settings, non-pathological circumstances (for example measurement noise) can severely restrict its efficiency. This work presents a complementary approach where one conditions on carefully designed linear combinations of data, which is particularly effective in the setting of jointly predicting many values in large connected regions of the data domain. For kernel functions that are smooth away from the origin and simple prediction domains, this method can be exponentially convergent in the number of linear combinations $r$ used for conditioning, and can be machine-precision machine-precision accurate for $r \approx 100$. This approach costs $\mathcal{O}(T r^2)$ work to compute where $T$ is the cost of solving a linear system with the data covariance matrix, and so in many cases can be computed in linear or near-linear cost by exploiting rank structure in well-behaved covariance matrices. At the cost of $\mathcal{O}(nr^2)$ additional precomputation work, this approach can also provide predictions at arbitrary points of a designated region in $\mathcal{O}(1)$ online work, making it particularly attractive for problems where prediction points are not known in advance.

翻译：高斯过程（GP）模型为预测与不确定性量化提供了灵活的框架。然而对于大多数协方差函数，基于n个点的精确GP预测计算复杂度为$\mathcal{O}(n^3)$，这使得其在处理大规模数据集或大量预测点时成本过高。虽然基于最近邻的预测在某些场景下表现良好，但非病态情形（如测量噪声）会严重限制其效率。本文提出了一种互补方法——通过精心设计的数据线性组合进行条件化，该方法在数据域大联通区域内联合预测多个值时尤为有效。对于原点处光滑的核函数及简单预测域，该方法在用于条件化的线性组合数r上可实现指数收敛，且当$r \approx 100$时可达到机器精度。该方法的计算量为$\mathcal{O}(T r^2)$，其中T为求解数据协方差矩阵线性系统的成本，因此通过利用优良协方差矩阵的秩结构，可在线性或近线性成本下实现计算。通过额外$\mathcal{O}(nr^2)$的预计算，该方法还能以$\mathcal{O}(1)$的在线计算成本提供指定区域内任意点的预测，尤其适用于预测点事先未知的问题。