高维复杂结构数据降维

With the development of technology, it is increasingly common to model the high dimensional data of complex structure. Modeling the matrix (tensor) valued data is commonly encountered in the field of medicine and biology. Although there are many theories and methods developed to deal with the high dimensional vector-valued data, vectorizing the matrix (tensor) data into vector simply and applying the existing methods for vector valued data will destroy the inner structure of the data and induce too many redundent paprameters, leading to unstable estimate. Compared with vector-valued case, parameters in the statistical model of matrix(tensor) data usally have more complicated structure. And the researches are less developed for many cases. This project will develop dimension reduction and varaible selection techniques for the high dimensional matrix(tensor) data of complex structure. It contains three parts. (1) Parameter estimation and hypothesis testing in the regression model of high dimensional matrix (tensor) valued data. （2）Robust statistical methods for high dimensional matrix(tensor) valued data. (3) The covariance and precision matrix estimate for high dimensional matrix（tensor） valued data. This project has great academic values and promising future in application.

随着技术进步，高维复杂结构数据的建模已经越来越普遍。在生物、医学等领域中经常需要对矩阵(张量)值(即变量取值为矩阵或张量)高维数据建模。尽管向量值（即变量的取值为向量）高维数据已有许多降维和变量选择方法，但是简单将矩阵(张量)值数据拉直为向量，并使用已有的向量值数据统计方法，将破坏数据的行列结构，导致参数维数过高，估计不稳定。和向量值数据相比，高维矩阵(张量)值数据建模中，参数往往具有更复杂的结构，而许多情形的研究还很不充分。本项目将研究具有复杂结构的高维矩阵(张量)数据的降维和变量选择方法，主要包括三个内容:(1)高维矩阵（张量）值数据回归模型中的参数估计和假设检验。(2)高维矩阵(张量)值数据的稳健统计方法。(3)高维矩阵(张量)值变量的协方差及其逆矩阵的估计。本项研究具有很高的学术价值和很强的应用价值。