Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics
Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics
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DOI：
10.48550/arxiv.2301.06533
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发表时间：
2023-01
期刊：
影响因子：
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通讯作者：
Mu Niu;Zhenwen Dai;P. Cheung;Yizhu Wang
中科院分区：
文献类型：
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作者：
Mu Niu;Zhenwen Dai;P. Cheung;Yizhu Wang
提出了一种构造点云中未知流形回归的本征高斯过程的新方法。在许多实际应用中，人们经常会遇到以一些低维未知流形为中心的高维数据(如点云数据)。流形的几何一般不同于通常的欧几里德几何。幼稚地将欧几里得高斯过程(GP)等传统平滑方法应用于多值数据，从而忽略空间的几何形状，可能会导致高度误导的预测和推断。嵌入高维欧氏空间的流形可以用概率映射函数及其对应的潜在空间来很好地描述。利用贝叶斯-高斯过程潜变量模型(BGPLVM)和黎曼几何研究了未知流形的几何结构。度量张量的分布是使用BGPLVM学习的。基于映射的不确定性量化来定义所得到的流形的边界。我们使用概率度量张量来模拟未知流形上的布朗运动路径。将热核估计为布朗运动的转移密度，并将其作为广义概率论的协方差函数。通过对瑞士辊、WiFi信号高维真实数据集和图像数据实例的仿真研究，说明了GPUM的应用。将其性能与图拉普拉斯GP、图材料GP和欧几里得GP进行了比较。
This article presents a novel approach to construct Intrinsic Gaussian Processes for regression on unknown manifolds with probabilistic metrics (GPUM) in point clouds. In many real world applications, one often encounters high dimensional data (e.g. point cloud data) centred around some lower dimensional unknown manifolds. The geometry of manifold is in general different from the usual Euclidean geometry. Naively applying traditional smoothing methods such as Euclidean Gaussian Processes (GPs) to manifold valued data and so ignoring the geometry of the space can potentially lead to highly misleading predictions and inferences. A manifold embedded in a high dimensional Euclidean space can be well described by a probabilistic mapping function and the corresponding latent space. We investigate the geometrical structure of the unknown manifolds using the Bayesian Gaussian Processes latent variable models(BGPLVM) and Riemannian geometry. The distribution of the metric tensor is learned using BGPLVM. The boundary of the resulting manifold is defined based on the uncertainty quantification of the mapping. We use the the probabilistic metric tensor to simulate Brownian Motion paths on the unknown manifold. The heat kernel is estimated as the transition density of Brownian Motion and used as the covariance functions of GPUM. The applications of GPUM are illustrated in the simulation studies on the Swiss roll, high dimensional real datasets of WiFi signals and image data examples. Its performance is compared with the Graph Laplacian GP, Graph Matern GP and Euclidean GP.