Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics
Mu Niu, Zhenwen Dai, Pokman Cheung, Yizhu Wang; 24(104):1−42, 2023.
Abstract
This article presents a new approach for constructing Intrinsic Gaussian Processes for regression on unknown manifolds with probabilistic metrics (GPUM) in point clouds. In many real-world applications, high-dimensional data (such as “point cloud data”) is often centered around lower-dimensional unknown manifolds. The geometry of these manifolds differs from the usual Euclidean geometry. Using traditional smoothing methods like Euclidean Gaussian Processes (GPs) on manifold-valued data without considering the geometry of the space can result in inaccurate predictions and inferences. To address this, we propose a probabilistic mapping function and corresponding latent space to describe a manifold embedded in a high-dimensional Euclidean space. We analyze the geometrical structure of the unknown manifolds using Bayesian Gaussian Processes latent variable models (B-GPLVM) and Riemannian geometry. The distribution of the metric tensor is learned using B-GPLVM, and the boundary of the resulting manifold is determined based on the uncertainty quantification of the mapping. We utilize the probabilistic metric tensor to simulate Brownian Motion paths on the unknown manifold and estimate the heat kernel as the transition density of Brownian Motion. This heat kernel is then used as the covariance function for GPUM. We demonstrate the effectiveness of GPUM through simulation studies on the Swiss roll, high-dimensional real datasets of WiFi signals, and image data examples. We also compare its performance with other methods such as Graph Laplacian GP, Graph Matérn GP, and Euclidean GP.
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