Tensor and Subspace Learning Methods with Applications to Medical Imaging

医学成像技术在临床和研究中都变得越来越重要。例如，锥形束计算机断层扫描在牙科诊断和治疗计划中发挥着重要作用，功能磁共振成像通常用于帮助研究人员确定大脑活动的特征。需要更有效和更强大的统计技术来分析医学成像，以诊断和预测疾病和其他疾病。通过这个项目，将开发新的统计理论、方法和算法来分析大型复杂数据，如医学成像数据集。这项研究有望提供新的理论见解和实用方法，推动多元统计学的发展，同时应对医学成像数据分析日益增长的需求和挑战。该项目的研究成果将通过与神经学家和生物医学工程师的合作以及大量研究生培训和外联活动来传播。现代科学和工程数据中嵌入了超高维、张量结构和高相关性。如果估计和推理技术忽略了变量之间的高度相关性、额外协变量引起的异质性或内在结构信息，则会变得低效甚至不一致。为了应对这些挑战，将开发统计严谨和计算效率高的学习方法。一个关键的思想是构造和估计目标子空间，以排除数据集中的噪声和无关信息。该研究项目预计将在两个方面做出重大贡献：高维包络子空间估计的计算和理论基础，以及回归和分类模型中的有效张量参数估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Medical imaging techniques have become increasingly important for both clinical and research studies. For example, cone beam computed tomography has a major role in diagnosis and treatment planning in dentistry, and functional magnetic resonance imaging is routinely used to help researchers characterize brain activity. More efficient and powerful statistical techniques are needed to analyze medical imaging for diagnosis and prediction of diseases and other disorders. Through this project, new statistical theory, methods, and algorithms will be developed for the analysis of large complex data such as medical imaging data sets. The research is expected to provide new theoretical insights and practical methodologies that advance multivariate statistics while simultaneously responding to the growing needs and challenges of medical imaging data analysis. Results of the research in this project will be disseminated through collaborations with neuroscientists and biomedical engineers, as well as substantial graduate student training and outreach activities. Open source and user-friendly software will also be produced.Ultra-high dimensionality, tensor structure, and high correlations are embedded in modern scientific and engineering data. Estimation and inferential techniques become inefficient or even inconsistent if they ignore high correlations among variables, heterogeneity caused by additional covariates, or intrinsic structural information. To address these challenges, statistically rigorous and computationally efficient learning methods will be developed. A key idea is to construct and estimate targeted subspaces that exclude the noise and irrelevant information in the data set. The research project is expected to make significant contributions on two fronts: computational and theoretical foundations for envelope subspace estimation in high dimensions, and efficient tensorial parameter estimation in regression and classification models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.