Large Matrix Estimation for Super-High Dimensional Data

为支持现代科学研究而生成的数据的数量和复杂性继续快速增长。大型数据集的特点是有许多变量或特征和/或许多样本，现在通常在从金融和生物医学科学到地球科学和工程等领域进行研究。如此大的复杂数据带来了许多统计和计算挑战，这些挑战是更传统的统计工具所不存在的，这些工具要求样本量远大于特征或变量的数量。与此同时，它们也为统计学科带来了前所未有的机遇。对于这些超高维数据，具有严格建立的属性的实用统计方法虽然仍然很困难，但对于许多前沿科学研究，如气候建模，投资组合分配和风险管理，量子计算和量子通信，基因表达研究和图像理解变得比以往任何时候都重要。该项目研究（i）大协方差矩阵的估计;（ii）高频金融中的大波动率矩阵;（iii）量子信息科学中的大密度矩阵。研究者试图通过稀疏性理论为基于复杂超高维数据的大型矩阵推理问题提供新的统计方法和理论。数字革命对科学研究和知识发现中的数据收集产生了深刻影响，技术进步使以相对较低的成本收集数据成为可能。 因此，为支持现代科学研究而生成的数据的数量和复杂性继续快速增长。大型数据集现在普遍用于从金融和生物医学科学到地球科学和工程等领域。如此大规模的复杂数据带来了许多传统统计工具所不具备的统计和计算挑战。与此同时，它们为统计提供了前所未有的机会。对于这些数据集，有效的统计方法对于许多前沿科学研究变得比以往任何时候都更加重要，如气候建模，投资组合分配和风险管理，量子计算和量子通信，基因表达研究和图像理解。该研究项目为分析如此庞大复杂的数据创造了先进有效的统计工具。研究者积极参与活动，将研究与学生培训相结合，并解决生物医学科学，地球科学，金融和量子信息科学领域的应用。

The amount and complexity of data generated to support modern scientific studies continues to grow rapidly. Large data sets, characterized by many variables or features and/or many samples, are now commonly studied in fields ranging from finance and biomedical sciences to geoscience and engineering. Such large complex data pose a number of statistical and computational challenges that are absent in more traditional statistical tools, where sample size is required to be much larger than the number of features or variables. At the same time, they present unprecedented opportunities to statistics discipline. For these super-high dimensional data, practical statistical methods with rigorously-established properties, while remain difficult, become more important than ever to many frontier scientific studies like climate modeling, portfolio allocation and risk management, quantum computation and quantum communication, gene expression study, and image understanding. This project studies the estimation of (i) large covariance matrices; (ii) large volatility matrices in high-frequency finance; (iii) large density matrices in quantum information science. The investigator intends to develop novel statistical methodologies and theories via sparsity for the large matrix inference problems based on complex super-high dimensional data. The research project has great potential to make a significant impact on the broad scientific community.Digital revolution has a profound impact on data collections in scientific research and knowledge discovery, and technological advances make it possible to collect data with relatively low costs. As a result, the amount and complexity of data generated to support modern scientific studies continues to grow rapidly. Large data sets are now commonly used in fields ranging from finance and biomedical sciences to geoscience and engineering. Such large scale, complex data pose a number of statistical and computational challenges that are absent in more traditional statistical tools. At the same time, they present unprecedented opportunities to statistics. For these data sets, valid statistical methods become more important than ever to many frontier scientific studies like climate modeling, portfolio allocation and risk management, quantum computation and quantum communication, gene expression study, and image understanding. The research project creates advanced effective statistical tools for the analysis of such vast complex data. The investigator actively engages in activities to integrate research with student training and address applications in the fields of biomedical sciences, geoscience, finance, and quantum information science.