CAREER: Utilizing Geometry for Statistical Learning and Inference

该项目的目标是研究几何在统计学中的基本作用，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以一些低维几何空间为中心。该计划的中心主题是几何本身存在于已知或待学习的几何数据中，应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方面取得根本性的数学、统计和算法进步。此外，研究者还为研究生、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，该计划与研究计划相结合。在许多科学领域中，经常收集复杂性质的现代数据。神经成像中的弥散张量成像（diffusion tensor imaging， DTI）就是一个例子，它通过3 × 3正定矩阵获得神经活动的局部信息。DTI在精神分裂症等神经系统疾病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形成特定形状的地标表示。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。研究者将描述复杂数据的结构或几何形状，并将几何形状纳入开发有效的统计模型进行推理。除了复杂的数据外，在生物学、公共卫生和神经科学等许多学科中收集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社会中准确的统计推断和决策至关重要。研究者将利用几何学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，在应用领域具有潜在的深远影响。特别是，该项目的统计方法的实际影响将在阿尔茨海默病神经影像学倡议（ADNI）数据库和注意缺陷多动障碍（ADHD）数据集的背景下进行评估。通过完成该计划，研究者希望将开发的模型和方法应用于医学诊断等重要领域，通过准确预测，分类或检测疾病或脑部疾病，更好地服务社会和推进科学。

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.