Nonparametric statistics on implicit manifolds learned via variational autoencoder

具有高维预测因子的非参数回归问题越来越受到人们的关注。在从计算机科学到环境科学的各种领域中，人们经常遇到被高维噪声干扰但以一些低维隐式流形为中心的高维数据(例如，点云数据)。流形的几何一般不同于通常的欧几里得几何。天真地将传统的多元分析应用于忽略空间几何的多值数据，可能会导致高度误导的预测和推断。牛等人。(2019)提出了复域上固有高斯过程(In-GP)的非参数光滑化方法。然而，对于大多数现实世界的问题，点云中的数据通常是高维的，不能在流形上直接观察到。在这个项目中，我们将使用深度生成建模方法(例如变分自动编码器)来估计隐式流形的概率参数化。利用黎曼几何研究了隐式流形的几何结构，并估计了度量张量。该方案的目的是通过在隐式流形上构造In-GP来填补高维点云中未定义流形在模型结构和推理方面的一个关键空白。

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.