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.
翻译:文章展示了一种新颖的方法, 用于在点云中构建具有概率度量度的未知元体回归的 Intrinsici Gausian 进程。 在许多现实世界应用中, 人们经常遇到以一些低维未知元体为中心的高维数据( 例如点云数据 ) 。 多元的几何与通常的 Euclidean 几何测量法一般不同。 将欧洲clidean Gausian 进程( GPS) 等传统平滑方法应用于多重估值数据, 从而忽略空间的几何测量方法, 可能会导致高度误导预测和推断。 在高维的 Euclidean 空间中嵌入的多维数据( 例如点云云云数据数据数据), 我们使用 Bayesian Gausial 进程潜伏模型( BGGPLVM) 和 Riemannian 几何测测算。 由此得出的多维度值值值的边界, 以高维值模型模型模型的精确度分析模型为模型。 我们使用不解的精确度测测测测测测测测测的轨道的轨道数据。