Gaussian processes are flexible probabilistic regression models which are
widely used in statistics and machine learning. However, a drawback is their
limited scalability to large data sets. To alleviate this, we consider
full-scale approximations (FSAs) that combine predictive process methods and
covariance tapering, thus approximating both global and local structures. We
show how iterative methods can be used to reduce the computational costs for
calculating likelihoods, gradients, and predictive distributions with FSAs. We
introduce a novel preconditioner and show that it accelerates the conjugate
gradient method's convergence speed and mitigates its sensitivity with respect
to the FSA parameters and the eigenvalue structure of the original covariance
matrix, and we demonstrate empirically that it outperforms a state-of-the-art
pivoted Cholesky preconditioner. Further, we present a novel, accurate, and
fast way to calculate predictive variances relying on stochastic estimations
and iterative methods. In both simulated and real-world data experiments, we
find that our proposed methodology achieves the same accuracy as Cholesky-based
computations with a substantial reduction in computational time. Finally, we
also compare different approaches for determining inducing points in predictive
process and FSA models. All methods are implemented in a free C++ software
library with high-level Python and R packages.

高斯过程是灵活的概率回归模型，但其计算规模受限；本文提出了全尺度近似方法，通过预测过程和协方差截尾相结合，减少计算开销，并引入新的预处理器和迭代方法以提高计算速度和预测方法准确度，在实验中证明它相较于现有方法在减少计算时间的同时，具备相同的准确性；此外，也比较了确定预测过程和全尺度近似模型中诱导点的不同方法。