Collaborative Research: Tensor Envelope Model - A New Approach for Regressions with Tensor Data

现代科学中最令人感兴趣的问题之一是了解人类的大脑。特别是，科学家们想了解患有神经系统疾病的人和没有神经系统疾病的人的大脑之间的差异。在大脑成像分析中，科学家以图像的形式收集数据，用于比较正常的衰老过程和神经系统疾病的发展。通过这个项目，PI寻求开发一个工具包，包括一套新颖的统计方法，理论和算法，用于分析大脑成像数据，以及各种科学和商业领域出现的类似数据。拟议的研究计划预计将在两个方面做出重大贡献：及时响应阵列数据分析日益增长的需求和挑战，并提供一类相关的方法，推进统计学科。本研究计划中的研究将通过研究人员与神经科学家的密切合作以及大量的教育和推广活动进行传播。多维数组或张量数据现在经常出现在广泛的科学和商业领域。针对张量数据分析中一些最紧迫的问题，本研究将先进的统计建模设备与现代计算技术相结合，开发一套新颖的张量回归方法。虽然已经有大量的文献对高维回归分析，几乎所有的工作是与向量响应或预测。将张量简单地转化为向量会导致维度的增加，破坏张量中固有的结构信息，并且通常会使经典方法变得不充分。本研究将开发用于张量响应或预测因子的回归建模的方法和工具，这些方法和工具既有效地解决了高维问题，又同时保留了张量结构。将研究三组问题：（1）具有包络的张量响应回归，旨在解决诸如在控制一组潜在的混杂变量之后识别疾病组和普通人群之间表现出不同活动模式的脑区域的问题;（2）带包络的张量预测回归，针对使用脑图像诊断神经变性疾病和预测神经精神疾病发作的问题;（3）带包络的协方差矩阵反应回归，旨在了解脑网络的变化并建立其与病理表型的关联。所有这些目标的核心思想是通过广义稀疏性原则和一类张量包络方法的发展。经典的稀疏性原理假设个体变量的子集是不相关的，并且采用各种惩罚函数来诱导这种稀疏性。相比之下，这种广义稀疏性原则假设变量的线性组合是不相关的，所提出的包络方法同时识别和排除这些不相关的信息，以实现更高的估计精度和效率。

One of the most intriguing questions in modern science is to understand the human brain. In particular, scientists want to understand the differences between the brains of people with neurological disorders and those without. In brain imaging analysis, scientists collect data in the form of images that are used to compare the normal aging process to the development of neurological disorders. Through this project, the PIs seek to develop a toolkit comprised of a set of novel statistical methods, theories, and algorithms for the analysis of brain imaging data, as well as similar data that arise in a variety of scientific and business fields. The proposed research program is expected to make significant contributions on two fronts: timely responding to the growing needs and challenges of array data analysis, and providing a class of associated methodology that advances the statistical discipline. Research proposed in this project is to be disseminated through the investigators' close collaborations with the neuroscientists, as well as substantial educational and outreach activities.Multidimensional array, or tensor, data are now frequently arising in a wide range of scientific and business fields. Aiming to address some of the most pressing questions in tensor data analysis, this research will integrate advanced statistical modeling devices with modern computational techniques to develop a set of novel tensor regression methods. Whereas there has been an enormous body of literature on high-dimensional regression analysis, nearly all work is with a vector response or predictor. Naively turning a tensor into a vector would result in ultrahigh dimensionality, destroy inherent structural information embedded in the tensor, and often render classical methods inadequate. This research will develop methods and tools for regression modeling of tensor responses or predictors, which both effectively tackles the high dimensionality and simultaneously preserves the tensor structure. Three sets of problems are to be investigated: (1) tensor response regression with envelope, aiming to address questions such as identifying brain regions exhibiting different activity patterns between the disease group and the general population after controlling for a set of potential confounding variables; (2) tensor predictor regression with envelope, aiming at questions of using brain images to diagnose neurodegenerative disorders and to predict onset of neuropsychiatric diseases; and (3) covariance matrix response regression with envelope, aiming to understand brain network alternations and building their associations with pathological phenotypes. The core idea underlying all of these aims is the adoption of a generalized sparsity principle and the development of a class of tensor envelope methods. The classical sparsity principle assumes a subset of individual variables are irrelevant, and various penalty functions are employed to induce such sparsity. By contrast, this generalized sparsity principle assumes linear combinations of variables are irrelevant, and the proposed envelope methods simultaneously identify and exclude such irrelevant information to achieve much improved estimation accuracy and efficiency.