A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information
A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information
复制标题
用函数信息对高斯过程建模的基于内核的方法
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
D. Brown
中科院分区：
文献类型：
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作者：
J. Nicholson;P. Kiessler;D. Brown
在机器学习和统计学中，高斯过程是建模连续过程最有用的工具之一。如果一个过程的值在一个有限的点集合上是已知的，则可以使用高斯过程来构造一个曲面，该曲面对这些值进行内插，以用于其他位置的预测和不确定性量化。然而，可用的信息并不总是以有限的点集合的形式存在。例如，边值问题包含关于域的边界的信息，域是不能合并到典型的高斯过程技术中的不可计数的点的集合。本文利用条件期望与正交投影的等价性，构造了一个利用再生核Hilbert空间将典型的有限情形与信息不可数的情形统一起来的高斯过程模型。我们在统计模型中讨论这种结构，包括数值考虑和概念证明。
Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to be used for prediction and uncertainty quantification in other locations. However, it is not always the case that the available information is in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, which is an uncountable collection of points that cannot be incorporated into typical Gaussian process techniques. In this paper we construct a Gaussian process model which utilizes reproducing kernel Hilbert spaces to unify the typical finite case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections. We discuss this construction in statistical models, including numerical considerations and a proof of concept.