Gaussian process is one of the most popular non-parametric Bayesian methodologies for modeling the regression problem. It is completely determined by its mean and covariance functions. And its linear property makes it relatively straightforward to solve the prediction problem. Although Gaussian process has been successfully applied in many fields, it is still not enough to deal with physical systems that satisfy inequality constraints. This issue has been addressed by the so-called constrained Gaussian process in recent years. In this paper, we extend the core ideas of constrained Gaussian process. According to the range of training or test data, we redefine the hat basis functions mentioned in the constrained Gaussian process. Based on hat basis functions, we propose a new sparse Gaussian process method to solve the unconstrained regression problem. Similar to the exact Gaussian process and Gaussian process with Fully Independent Training Conditional approximation, our method obtains satisfactory approximate results on open-source datasets or analytical functions. In terms of performance, the proposed method reduces the overall computational complexity from $O(n^{3})$ computation in exact Gaussian process to $O(nm^{2})$ with $m$ hat basis functions and $n$ training data points.

本文扩展了约束高斯过程的核心思想，并提出了一种新的基于“帽子”基函数的稀疏高斯过程方法，将计算复杂度从精确高斯过程的 $O(n^{3})$ 降低到 $O(nm^{2})$，在开源数据集或分析函数上取得了满意的近似结果。

基于帽子基函数的稀疏高斯过程