Inference for Gaussian Processes with Matérn Covariogram on Compact Riemannian Manifolds.

IF 4.3 3区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS
Journal of Machine Learning Research Pub Date : 2023-03-01
Didong Li, Wenpin Tang, Sudipto Banerjee
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引用次数: 0

Abstract

Gaussian processes are widely employed as versatile modelling and predictive tools in spatial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Matérn covariogram and heat kernel, the behaviour of asymptotic inference on the parameters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Building upon a recently introduced Matérn covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum likelihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory.

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紧凑黎曼曼形上具有马特恩协方差的高斯过程推理
高斯过程是空间统计学、函数数据分析、计算机建模和机器学习各种应用中广泛使用的通用建模和预测工具。人们对欧几里得空间上的高斯过程进行了广泛的研究,利用协方差函数或协方差图对复杂的依赖关系进行建模。关于黎曼流形上的高斯过程的文献越来越多,以便为非欧几里得数据开发更丰富、更灵活的推理框架。虽然通过图形表示对马特恩协方差和热核的数值近似进行了深入研究,但对协方差参数的渐近推断行为的关注却相对较少。我们重点研究在紧凑黎曼流形上构建的高斯过程的渐近行为。以最近引入的紧凑黎曼流形上的马特恩协变图为基础,我们采用紧凑流形上两个马特恩高斯随机度量等价的形式化概念和条件,推导出可识别的参数(也称为微角参数),并正式建立最大似然估计的一致性和最佳线性无偏预测器的渐近最优性。我们将圆作为紧凑黎曼流形的一个具体实例进行研究,并通过数值实验来说明和证实这一理论。
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来源期刊
Journal of Machine Learning Research
Journal of Machine Learning Research 工程技术-计算机:人工智能
CiteScore
18.80
自引率
0.00%
发文量
2
审稿时长
3 months
期刊介绍: The Journal of Machine Learning Research (JMLR) provides an international forum for the electronic and paper publication of high-quality scholarly articles in all areas of machine learning. All published papers are freely available online. JMLR has a commitment to rigorous yet rapid reviewing. JMLR seeks previously unpublished papers on machine learning that contain: new principled algorithms with sound empirical validation, and with justification of theoretical, psychological, or biological nature; experimental and/or theoretical studies yielding new insight into the design and behavior of learning in intelligent systems; accounts of applications of existing techniques that shed light on the strengths and weaknesses of the methods; formalization of new learning tasks (e.g., in the context of new applications) and of methods for assessing performance on those tasks; development of new analytical frameworks that advance theoretical studies of practical learning methods; computational models of data from natural learning systems at the behavioral or neural level; or extremely well-written surveys of existing work.
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