Extrinsic Gaussian Processes for Regression and Classification on Manifolds
Extrinsic Gaussian Processes for Regression and Classification on Manifolds
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DOI：
10.1214/18-ba1135
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发表时间：
2017-06
影响因子：
4.4
通讯作者：
Lizhen Lin;Mu Niu;P. Cheung;D. Dunson
中科院分区：
文献类型：
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作者：
Lizhen Lin;Mu Niu;P. Cheung;D. Dunson
高斯过程（GPs）在从回归到分类到空间过程的各种应用中被广泛用于未知函数或曲面的建模。尽管关于GPs的应用、方法、理论和算法的文献越来越多，但绝大多数文献都集中在输入域对应欧几里得空间的情况下。然而，特别是近年来随着复杂数据的收集越来越多，输入域通常不具有如此简单的形式。例如，输入通常被限制为非欧几里得流形，这种情况形成了本文的动机。特别地，我们提出了流形上GP建模的一般外在框架，该框架依赖于流形嵌入到欧几里德空间中，然后在其图像上构造GP的外在核。这些外在高斯过程（eGPs）被用作贝叶斯推理中未知函数的先验分布。我们的方法是简单和一般的，我们证明了egp继承了欧几里得空间中GP模型的优良理论性质。我们考虑了我们的模型在回归和分类问题中的应用，预测器位于一大类流形中，包括球体，平面形状空间，正定矩阵空间和格拉斯曼。我们的模型可以很容易地被生物科学从业者用于各种回归和分类问题，例如疾病诊断或检测。当空间位置在球体或其他几何空间上时，我们的工作也可能对空间统计产生影响。
Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians. Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces.