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This article presents a novel approach to construct Intrinsic GaussianProcesses for regression on unknown manifolds with probabilistic metrics (GPUM)in point clouds. In many real world applications, one often encounters highdimensional data (e.g. point cloud data) centred around some lower dimensionalunknown manifolds. The geometry of manifold is in general different from theusual Euclidean geometry. Naively applying traditional smoothing methods suchas Euclidean Gaussian Processes (GPs) to manifold valued data and so ignoringthe geometry of the space can potentially lead to highly misleading predictionsand inferences. A manifold embedded in a high dimensional Euclidean space canbe well described by a probabilistic mapping function and the correspondinglatent space. We investigate the geometrical structure of the unknown manifoldsusing the Bayesian Gaussian Processes latent variable models(BGPLVM) andRiemannian geometry. The distribution of the metric tensor is learned usingBGPLVM. The boundary of the resulting manifold is defined based on theuncertainty quantification of the mapping. We use the the probabilistic metrictensor to simulate Brownian Motion paths on the unknown manifold. The heatkernel is estimated as the transition density of Brownian Motion and used asthe covariance functions of GPUM. The applications of GPUM are illustrated inthe simulation studies on the Swiss roll, high dimensional real datasets ofWiFi signals and image data examples. Its performance is compared with theGraph Laplacian GP, Graph Matern GP and Euclidean GP.

Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics