Statistical Analysis of High Dimensional Manifold Data

多值数据经常出现在形状和图像分析、计算机视觉、生物力学等许多领域。人体器官形状的可变性研究中的医学成像数据位于相当高维的非线性流形上，其中的挑战是双重的：高维和低样本量(收集数据的成本很高)和自然施加的非欧几里德几何。该研究旨在回答物体形状分析中出现的科学问题，并为更复杂的问题提供坚实的数学基础。首先，研究者采用并扩展了正则化框架下的后向降维策略。提出了形状数据和方向数据的稀疏表示方法，并研究了它们的性质。在回归的背景下，提出了模型和检验非测地线趋势的流形响应的多项式回归。文中还考虑了局部多项式建模的推广。研究人员还探索了有效的计算方法。对物体形状的研究对于了解人体解剖物体的群体以及揭示生物标志物/临床结果和物体形状变化之间的相互作用至关重要。由于技术的先进性，现代物体形状数据变得庞大和复杂，但传统的方法缺乏对数据类型特殊几何结构的考虑。本研究项目旨在为包括物体形状在内的大规模非标准数据类型的探索性和验证性分析提供新的统计方法。该项目还将生产统计工具，这些工具可能会应用于许多领域，包括生物力学、计算机视觉、医学研究和生物科学。

Manifold-valued data appear frequently in shape and image analysis, computer vision, biomechanics and many others. Medical imaging data in studies of the variability of human organ shapes lie on a fairly high dimensional nonlinear manifolds, where the challenge is two-fold: high dimensionality with a low sample size (data are expensive to gather) and naturally imposed non-Euclidean geometry. The proposed research aims to answer scientific questions arising in object shape analysis and to provide solid mathematical basis for more complex problems. First, the investigator takes and extends the strategy of backward dimension reduction with regularization framework. Sparse representation of shape and directional data are proposed and their properties are studied. In the regression context, polynomial regression for manifold-valued response to model and test non-geodesic trends is proposed. An extension of local polynomial modeling is also considered. The investigator also explores efficient computational methods. The study of object shape is crucial for understanding the population of human anatomical objects and revealing the interplay between biomarkers/clinical outcomes and object shape variations. Due to the advanced technology, the modern object shape data become big and complex, but conventional methods lack considerations on the special geometric structure of the data types. This research project aims to provide new statistical methodologies for exploratory and confirmatory analysis of the large-scale non-standard data types, including the object shapes. The project will also produce statistical tools, which may be applied in many fields including biomechanics, computer vision, medical studies and biological sciences.