CAREER: Utilizing Geometry for Statistical Learning and Inference

该项目的目标是研究几何在统计中的基本作用，并利用它进行学习和推理。 更具体地说，研究者建议（1）研究几何在复杂数据的统计推断中的作用，特别是现在在许多科学和工程领域中常规收集的流形值数据;（2）研究几何在高维数据分析中的作用，其中数据生成过程通常围绕一些低维几何空间进行。 该程序的中心主题是几何形状固有地存在于数据中，其中几何形状是已知的或要学习的，应该用于有效和可靠的统计学习和推理。研究者的目标是在复杂数据分析和高维数据分析中从根本上提高数学，统计和算法的进步。此外，研究员还为研究生、本科生和高中生提出了一个全面而详细的教育和培训计划，并将其纳入研究计划。在许多科学领域，复杂性质的现代数据正在被定期收集。一个例子是来自神经成像的扩散张量成像（DTI），其通过3乘3正定矩阵获得神经活动的局部信息。DTI在神经系统疾病（如精神分裂症）的研究和治疗以及检测与各种疾病（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。复杂数据的其他示例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表示，形成某些形状。 还可能遇到以复杂形式存储的数据，例如子空间、曲面、曲线和网络。研究人员将描述复杂数据的结构或几何形状，并将几何形状纳入有效的统计模型中进行推理。除了复杂的数据，收集生物学、公共卫生和神经科学等多个学科的高维数据也是一种常见的做法。能够学习高维数据的低维几何对于准确的统计推断和社会决策至关重要。研究者将利用几何学来开发有效的统计方法，这些方法将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数据库和注意力缺陷多动障碍数据集的背景下评估该项目的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学诊断等重要领域，从而更好地服务于社会和推进科学，从而实现疾病或脑部疾病的准确预测，分类或检测。

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.