In order to perform GP predictions fast in large geospatial fields with small-scale variations, a computational complexity that is independent of the number of measurements $N$ and the size of the field is crucial. In this setting, GP approximations using $m$ basis functions requires $\mathcal{O}(Nm^2+m^3)$ computations. Using finite-support basis functions reduces the required number of computations to perform a single prediction to $\mathcal{O}(m^3)$, after a one-time training cost of $O(N)$. The prediction cost increases with increasing field size, as the number of required basis functions $m$ grows with the size of the field relative to the size of the spatial variations. To prevent the prediction speed from depending on field size, we propose leveraging the property that a subset of the trained system is a trained subset of the system to use only a local subset of $m'\ll m$ finite-support basis functions centered around each prediction point to perform predictions. Our proposed approximation requires $\mathcal{O}(m'^3)$ operations to perform each prediction after a one-time training cost of $\mathcal{O}(N)$. We show on real-life spatial data that our approach matches the prediction error of state-of-the-art methods and that it performs faster predictions, also compared to state-of-the-art approximations that lower the prediction cost of $\mathcal{O}(m^3)$ to $\mathcal{O}(m\log(m))$ using a conjugate gradient solver. Finally, we demonstrate that our approach can perform fast predictions on a global bathymetry dataset using millions of basis functions and tens of millions of measurements on a laptop computer.
翻译:为了在规模小且变化小的大地理空间域快速进行GP预测,一个与测量数量N美元和字段大小无关的计算复杂度至关重要。在这一背景下,使用美元基函数的GP近似值需要$\mathcal{O}(Nm2\2+m3}3美元计算。使用有限支持基础功能可以减少进行单一预测的计算数量,在一次性培训费用为$(N)美元之后,计算费用将增加。随着外地规模的扩大,预测费用将随着外地规模的增加而增加。在这种背景下,使用美元基函数的大小与空间变异的大小相比,需要美元基函数将增加$。为了防止预测速度取决于实地大小,我们建议利用经过训练的系统的一个子集来减少进行单项的计算数量,在每次预测点(美元)之后,我们拟议的近似需要美元基值的美元基数的预测成本(MAR_3) 。我们用一个更低的精确值的计算方法来进行一个更快速的运行。