Sparse Gaussian Process Based On Hat Basis Functions

TL;DR

This paper introduces a new sparse Gaussian process method using hat basis functions that efficiently handles regression problems, reducing computational complexity while maintaining accuracy, and extends constrained Gaussian process ideas.

Contribution

It extends constrained Gaussian process concepts by redefining hat basis functions based on data range and proposes a sparse Gaussian process method with lower computational complexity.

Findings

Achieves comparable results to exact Gaussian processes on benchmark datasets.

Reduces computational complexity from O(n^3) to O(nm^2).

Demonstrates effectiveness on analytical functions and open-source datasets.

Abstract

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.