CAREER: Utilizing Geometry for Statistical Learning and Inference

该项目的目标是研究几何学在统计学中的基本作用，并将其用于学习和推理。更具体地说，研究人员建议(1)研究几何在复杂数据的统计推断中的作用，特别是现在许多科学和工程领域中常规收集的流形值数据；以及(2)调查几何在高维数据分析中的作用，在高维数据分析中，数据生成过程通常围绕一些低维几何空间。本程序的中心主题是，几何图形固有地存在于数据中，几何图形既可以是已知的，也可以是要学习的，这应该被用于有效和可靠的统计学习和推理。研究人员的目标是在复杂数据分析和高维数据分析方面取得根本性的数学、统计和算法进步。此外，调查员还为研究生、本科生以及高中生提出了全面而详细的教育和培训计划，并将其纳入研究计划。许多科学领域经常收集复杂性质的现代数据。一个例子是神经成像的扩散张量成像(DTI)，它通过3×3的正定矩阵获得神经活动的局部信息。DTI在研究和治疗精神分裂症等神经疾病以及检测与多种疾病(包括中风、多发性硬化症、阅读障碍)相关的细微异常方面具有临床应用。复杂数据的其他例子包括机器视觉中的数字图像，其中数字图像可以由一组地标来表示，形成特定的形状。人们还可能遇到以复杂形式存储的数据，如子空间、曲面、曲线和网络。研究人员将描述复杂数据的结构或几何结构，并将几何结构纳入开发有效的统计模型以进行推理。除了复杂的数据外，收集生物学、公共卫生和神经科学等许多学科的高维数据也是一种常见的做法。能够学习高维数据往往是低维的几何图形，对于社会上准确的统计推断和决策是至关重要的。研究人员将利用几何学开发有效的统计方法，并将其应用于医学数据和神经科学数据，这在应用领域具有潜在的深远影响。特别是，将在阿尔茨海默病神经成像倡议(ADNI)数据库和注意力缺陷多动障碍(ADHD)数据集的背景下评估该项目统计方法的实际影响。通过完成拟议的计划，研究人员希望通过将开发的模型和方法应用于医疗诊断等重要领域，实现对疾病或大脑疾病的准确预测、分类或检测，从而更好地服务于社会和推动科学进步。

The goal of the project is to study the fundamental role of geometry in statistics and utilize it for learning and inference. More specifically, the investigator proposes to (1) study the role of geometry in statistical inference of complex data, in particular manifold-valued data that are now routinely collected in many fields of science and engineering; and (2) investigate the role of geometry in high-dimensional data analysis where the data generating process often centers around some lower-dimensional geometric space. The central theme of this program is that geometry is inherently present in the data with the geometry either known or to be learned, which should be utilized for efficient and reliable statistical learning and inference. The investigator aims to make fundamentally mathematical, statistical and algorithmic advances in complex data analysis and high-dimensional data analysis. In addition, the investigator proposes a comprehensive and detailed educational and training program for graduates students, undergraduate students as well as high school students that is integrated into the research program.Modern data of complex nature are routinely being collected in many scientific fields. One example is from diffusion tensor imaging (DTI) of neuroimaging, which obtains local information of neural activity through 3 by 3 positive definite matrices. DTI has clinical applications in the study and treatment of neurological disorders such as schizophrenia, as well as in detecting subtle abnormalities related to a variety of diseases (including stroke, multiple sclerosis, dyslexia). Other examples of complex data include digital images in machine vision, where a digital image can be represented by a set of landmarks, forming certain shapes. One may also encounter data that are stored in complex forms such as subspaces, surfaces, curves, and networks. The investigator will characterize the structure or geometry of complex data, and incorporate the geometry in developing valid statistical models for inference. In addition to complex data, it is also a common practice to collect high-dimensional data across many disciplines such as biology, public health and neuroscience. Being able to learn the often lower-dimensional geometry of the high-dimensional data is essential for accurate statistical inference and decision making in society. The investigator will utilize the geometry in developing valid statistical methods, which will be applied to medical data and neuroscience data, which has potential far-reaching impact in applied fields. In particular, the practical impact of the statistical methodologies from the project will be evaluated in the context of the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, and an Attention Deficit Hyperactivity Disorder (ADHD) data set. By completing the proposed program, the investigator expects to better serve society and advance science by applying the developed models and methods in important fields such as medical diagnostics by enabling accurate prediction, classification or detection of diseases or brain disorders.