Nonparametric statistics on implicit manifolds learned via variational autoencoder

There are increasing interests in the problem of nonparametric regression with high dimensional predictors. In a variety of fields, from computer science to environmental science, one often encounters high dimensional data (e.g., 'point cloud data') perturbed by high-dimensional noise but centering around some lower-dimensional implicit manifolds. The geometry of the manifold is in general different from the usual Euclidean geometry. Naively applying traditional multivariate analysis to manifold-valued data that ignores the geometry of the space can potentially leads to highly misleading predictions and inferences. Niu et al. (2019) proposed the nonparametric smoothing methods of the intrinsic Gaussian process (In-GP) on complex domains of which the geometry is known. However, for most of real-world problems, data in the point cloud, often high dimensional, is not directly observed on the manifold.

In this project we will estimate the probabilistic parameterization of the implicit manifolds using a deep generative modelling approach such as variational autoencoder. We investigate the geometrical structure of the implicit manifold using Riemannian geometry and estimate the metric tensor. The objective of this proposal is to fill a critical gap in model structure and inference for undefined manifolds in high dimension point clouds, by constructing the In-GP on implicit manifolds.