Nonparametric regression on implicit manifold of high dimensional point cloud

在从生物、生命科学到环境科学的各种领域中，人们经常会遇到高维数据(例如，点云数据)受到一些高维噪声的干扰，但却在一些低维流形上徘徊。在更精确的数学术语中，流形是具有某种微分/光滑结构的拓扑空间，其几何通常不同于通常的欧几里德几何。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。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。高斯过程(GP)是统计和机器学习中最强大的工具之一，用于回归和欧几里得预报器的优化。然而，如果预测器位于流形上，传统的平滑或建模方法(如GP)不考虑空间的固有几何和边界约束，结果会很差。在流形上发展GP模型的最大挑战之一是难以通过在流形上构造有效且可计算的协方差核来描述协方差结构。在Pi以前和最近的工作中，利用热核来构造复约束区域上的内禀高斯过程(In-GP)。虽然热核在理论上提供了一种自然和规范的选择，但直接评估在分析上是困难的。或者，它可以被估计为流形上布朗运动(BM)的转移密度。这项初步工作已被用于模拟咸海的叶绿素浓度水平。此方法仅适用于已知流形几何的情况。例如，从经度和纬度到R3中的球体的映射是已知的。然而，在现代大数据分析中，点云中的数据通常是高维的，不是直接在流形上观察到的，而是在潜在的高维环境空间中观察到的，但集中在一些未知的低维结构周围。人们需要先学习这个几何，然后才能利用它进行推理。该方案的目的是通过构造固有的高斯过程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。我们将使用贝叶斯降维方法来学习隐式流形。贝叶斯模型对于不确定性的量化非常重要。热核可以估计为BM在学习流形上的跃迁密度。这个框架将允许我们在点云上建立In-GP回归模型。点云上的本征GP可以充分结合学习流形的固有几何进行推理，同时考虑到潜在的复杂内部结构和边界。

In a variety of fields, from biology, life science, to environmental science, one often encounters high-dimensional data (e.g., 'point cloud data') perturbed by some high-dimensional noise but cantering around some lower-dimensional manifolds. In more precise mathematical terms, manifolds are topological spaces equipped with some differential/smooth structure, the geometry of which 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 lead to highly misleading predictions and inferences. There is increasing interest in the problem of nonparametric regression with high-dimensional predictors. Gaussian Processes (GP) are among the most powerful tools in statistics and machine learning for regression and optimisation with Euclidean predictors. However, if the predictors locate on a manifold, traditional smoothing or modelling methods like GP that do not respect the intrinsic geometry of the space and the boundary constraints, produce poor results. One of the paramount challenges in developing GP models on manifold is the difficulty in specifying the covariance structure via constructing valid and computable covariance kernels on manifolds. In PI's prior and recent work, the heat kernel is employed to construct the intrinsic Gaussian process (In-GP) on complex constrained domain. Although the heat kernel provides a natural and canonical choice theoretically, it is analytically intractable to directly evaluate. Alternatively, it can be estimated as the transition density of the Brownian Motion (BM) on the manifold. This preliminary work has been applied to model the chlorophyll concentration levels in Aral Sea. This method is only applicable when the geometry of the manifold is known. For example, the mapping from longitude and latitude to the sphere in R3 is known. However, in modern big data analysis, the data in the point cloud, often high dimensional, are not directly observed on the manifolds, instead they are observed in a potentially high-dimensional ambient space but concentrate around some unknown lower dimensional structure. One needs to learn this geometry before utilizing it for inference. 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 intrinsic Gaussian processes. We will use the Bayesian dimension reduction method to learn the implicit manifold. The Bayesian models are extremely important for uncertainty quantification. The heat kernel can be estimated as the transition density of the BM on the learned manifold. This framework will allow us to build the In-GP regression models on point cloud. The intrinsic GP on point cloud can incorporate fully the intrinsic geometry of the learned manifold for inference while respecting the potentially complex interior structure and boundary.