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High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps

机译：使用高斯进程回归和扩散图的高维内联插值

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摘要

This article considers the challenging task of estimating geologic properties of interest using a suite of proxy measurements. The current work recast this task as a manifold learning problem. In this process, this article introduces a novel regression procedure for intrinsic variables constrained onto a manifold embedded in an ambient space. The procedure is meant to sharpen high-dimensional interpolation by inferring non-linear correlations from the data being interpolated. The proposed approach augments manifold learning procedures with a Gaussian process regression. It first identifies, using diffusion maps, a low-dimensional manifold embedded in an ambient high-dimensional space associated with the data. It relies on the diffusion distance associated with this construction to define a distance function with which the data model is equipped. This distance metric function is then used to compute the correlation structure of a Gaussian process that describes the statistical dependence of quantities of interest in the high-dimensional ambient space. The proposed method is applicable to arbitrarily high-dimensional data sets. Here, it is applied to subsurface characterization using a suite of well log measurements. The predictions obtained in original, principal component, and diffusion space are compared using both qualitative and quantitative metrics. Considerable improvement in the prediction of the geological structural properties is observed with the proposed method.
机译：本文考虑使用套件代理测量估算利益地质特性的具有挑战性的任务。当前的工作重新重新将此任务重新定为歧管学习问题。在该过程中，本文介绍了一种新的回归过程，用于限制嵌入环境空间中的歧管上的内部变量。该过程旨在通过从内插的数据推断出非线性相关性来锐化高维插值。建议的方法增强了具有高斯过程回归的流形学习程序。它首先使用扩散图识别嵌入在与数据相关联的环境高维空间中的低维歧管。它依赖于与该结构相关联的扩散距离来定义配备数据模型的距离功能。然后，该距离度量函数用于计算高斯过程的相关结构，其描述高维环境空间中感兴趣量的统计依赖性。所提出的方法适用于任意高维数据集。这里，使用套件的日志测量套件应用于地下表征。使用定性和定量度量来比较原始，主成分和扩散空间中获得的预测。用该方法观察到地质结构特性预测的相当大的改进。

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