CIF: Small: Efficient Discriminant Analysis Through Parsimonious Probabilistic Models

在大数据时代，迫切需要开发更高效、更易于解释的数据分析方法，以应对来自现实生活应用的新挑战，如癌症诊断、脑连接网络和神经成像研究。除了它们令人望而生畏的大小之外，这类应用中的数据集还要求研究人员了解它们背后的内在和复杂结构。本研究通过探索最新的统计方法和现代计算技术，在判别分析和分类方面涉及到新的方向。研究人员将开发新的统计方法和计算工具，以利用数据的内在结构，如高相关性、网络结构和张量/数组数据结构。研究成果将推动统计学、机器学习、文本挖掘、生物医学研究、金融、神经成像分析等领域的研究。这项研究还将与大量的教育和推广活动相结合。在这项研究中，研究人员的目标是开发简约的概率判别分析模型，以有效地分析具有内在复杂结构的数据，并改进参数估计和预测的准确性。本文将研究三组问题：(1)带包络的简约线性判别分析，旨在将包络建模的新技术与经典线性判别分析模型相结合；(2)同时判别分析和差分网络估计，旨在为同时研究差分网络、估计多个协方差矩阵和训练二次判别分析分类器提供一个新的统一框架；(3)稀疏张量判别分析和特征选择，旨在发展一种针对张量值数据的稀疏判别分析方法，该方法直接利用张量值特征进行判别分析，同时实现特征选择和保留可解释张量结构。

In the Big-Data era, there is a pressing need to develop more efficient and easy-to-interpret data analysis methods to face the new challenges rising from real life applications, such as cancer diagnosis, brain connectivity network and neuroimaging studies. Besides their intimidating sizes, data sets in such applications demand researchers to understand the intrinsic and complex structure behind them. This research involves new directions in discriminant analysis and classification by exploring state-of-art statistical methods and contemporary computation techniques. The investigators will develop new statistical methods and computational tools to utilize the intrinsic structure of the data, such as high correlation, network structure and tensor/array data structure. The research results will advance the research in statistics, machine learning, text mining, bio-medical research, finance, neuroimaging analysis, among other fields. The research will also be integrated with substantial educational and outreach activities.In this research, the investigators aim to develop parsimonious probabilistic discriminant analysis models for efficiently analyzing data with intrinsically complicated structures, and for improved estimation of parameters and accuracy in predictions. Three sets of problems will be investigated: (1) parsimonious linear discriminant analysis with envelope, aiming to integrate a nascent technique of envelope modeling with the classical linear discriminant analysis model; (2) simultaneous discriminant analysis and differential network estimation, aiming to develop a novel and unified framework for simultaneously studying differential networks, estimating multiple covariance matrices, and training quadratic discriminant analysis classifier; and (3) sparse tensor discriminant analysis with feature selection, aiming to develop a sparse discriminant analysis method for tensor-valued data that directly uses the tensor-valued features for discriminant analysis, while simultaneously achieving feature selection and preserving interpretable tensor structure.