Nonparametric statistics on implicit manifolds learned via variational autoencoder

通过变分自动编码器学习的隐式流形的非参数统计

• 批准号: 2750752
• 金额: --
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2750752
• 关键词: Nonparametric; statistics; implicit; manifolds; learned

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.

高维预测变量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，人们经常遇到高维数据（例如，“点云数据”）受高维噪声扰动，但以一些低维隐式流形为中心。流形的几何一般不同于通常的欧几里得几何。简单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的预测和推断。Niu等人（2019）提出了几何形状已知的复杂域上的内在高斯过程（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，往往是高维的，不能直接观察到的流形上。 在这个项目中，我们将估计概率参数化的隐式流形使用深生成建模方法，如变分自动编码器。我们利用黎曼几何研究了隐流形的几何结构，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中的一个关键空白。