Among generalized additive models, additive Mat\'ern Gaussian Processes (GPs) are one of the most popular for scalable high-dimensional problems. Thanks to their additive structure and stochastic differential equation representation, back-fitting-based algorithms can reduce the time complexity of computing the posterior mean from $O(n^3)$ to $O(n\log n)$ time where $n$ is the data size. However, generalizing these algorithms to efficiently compute the posterior variance and maximum log-likelihood remains an open problem. In this study, we demonstrate that for Additive Mat\'ern GPs, not only the posterior mean, but also the posterior variance, log-likelihood, and gradient of these three functions can be represented by formulas involving only sparse matrices and sparse vectors. We show how to use these sparse formulas to generalize back-fitting-based algorithms to efficiently compute the posterior mean, posterior variance, log-likelihood, and gradient of these three functions for additive GPs, all in $O(n \log n)$ time. We apply our algorithms to Bayesian optimization and propose efficient algorithms for posterior updates, hyperparameters learning, and computations of the acquisition function and its gradient in Bayesian optimization. Given the posterior, our algorithms significantly reduce the time complexity of computing the acquisition function and its gradient from $O(n^2)$ to $O(\log n)$ for general learning rate, and even to $O(1)$ for small learning rate.

本文针对加性Matern高斯过程，提出基于后向逼近算法的计算方法，计算后验均值、方差、似然函数和梯度的复杂性从O(n^3)降低到O(nlogn)，并应用于贝叶斯优化中，提出了后验更新、超参数学习、习得函数和其梯度的有效算法。

用稀疏矩阵表示加性高斯过程