Nonparametric statistics on implicit manifolds learned via variational autoencoder

具有高维预测量的非参数回归问题引起了人们越来越多的兴趣。在各种领域，从计算机科学到环境科学，人们经常遇到高维数据（例如，“点云数据”）被高维噪声扰动，但围绕一些低维隐式流形。流形的几何一般不同于通常的欧几里得几何。天真地将传统的多变量分析应用于流形值数据，忽略了空间的几何形状，可能会导致高度误导性的预测和推断。Niu等人（2019）在几何形状已知的复杂域上提出了内禀高斯过程（In-GP）的非参数平滑方法。然而，对于大多数现实问题，点云中的数据通常是高维的，不能直接在流形上观察到。在这个项目中，我们将使用深度生成建模方法（如变分自编码器）估计隐流形的概率参数化。利用黎曼几何研究隐流形的几何结构，并估计其度量张量。本文的目标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的模型结构和推理的关键空白。

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.