Local, algebraic simplifications of Gaussian random fields

Local, algebraic simplifications of Gaussian random fields
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高斯随机场的局部代数简化

DOI:
10.1088/1475-7516/2018/12/022
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发表时间:
2018
影响因子:
6.4
通讯作者:
Bjorkmo T
Bjorkmo T
中科院分区:
物理与天体物理2区
文献类型:
--
作者:
Bjorkmo T

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高斯随机场和高斯随机过程的许多应用都受到评估概率密度函数的计算复杂性的限制,其中涉及求逆相关协方差矩阵。在这项工作中,我们展示了如何使用高斯(或平方指数)协方差函数来完全规避高斯随机场的局部泰勒系数的问题。我们的结果适用于场的任何维度和泰勒展开式中的任何阶。我们提出两个应用。首先,我们证明该方法可用于显式生成具有多个场的非平凡势能景观。当人们关注特殊点(例如最大值或最小值)周围的局部场时,该应用程序特别有用,正如我们通过早期宇宙中的宇宙多场膨胀问题来举例说明的那样。其次,我们证明了该方法在机器学习中的应用,并且在给定由单点局部泰勒系数组成的训练数据集的情况下,极大地简化了确定协方差函数超参数的回归问题。随附的 Mathematica 笔记本可在\href {https://doi. org/10.17863/CAM。 22859}{https://doi. org/10.17863/CAM。 22859}。
Many applications of Gaussian random fields and Gaussian random processes are limited by the computational complexity of evaluating the probability density function, which involves inverting the relevant covariance matrix. In this work, we show how that problem can be completely circumvented for the local Taylor coefficients of a Gaussian random field with a Gaussian (orsquare exponential') covariance function. Our results hold for any dimension of the field and to any order in the Taylor expansion. We present two applications. First, we show that this method can be used to explicitly generate non-trivial potential energy landscapes with many fields. This application is particularly useful when one is concerned with the field locally around special points (eg maxima or minima), as we exemplify by the problem of cosmicmanyfield'inflation in the early universe. Second, we show that this method has applications in machine learning, and greatly simplifies the regression problem of determining the hyperparameters of the covariance function given a training data set consisting of local Taylor coefficients at single point. An accompanying Mathematica notebook is available at\href {https://doi. org/10.17863/CAM. 22859}{https://doi. org/10.17863/CAM. 22859}.
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发表时间: --
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作者:
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DOI: 10.1088/1475-7516/2018/02/037
发表时间: 2018
影响因子: 6.4
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发表时间: 2016
影响因子: 8.6
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发表时间: 2011
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通讯作者: A. Liddle
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发表时间: 2018
影响因子: 5.4
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