High-dimensional statistical learning and inference

高维度的挑战是当代许多统计问题的特点，这些问题来自科学研究和技术发展的许多前沿领域。在高维统计研究中，需要探索低维结构，这种结构在适当的参数化下具有稀疏性，以避免噪声随维度累积的问题。这项提议意在面对基因组学、机器学习、健康研究、经济学和金融学中的一些重要的高维统计问题。这些问题包括微阵列数据分析中出现的各种问题，如归一化、显著性分析和疾病分类；从高维统计学习中选择变量和提取特征；从高维特征空间中进行稀疏分类和聚类；用于资产配置和投资组合管理的高维协方差矩阵估计；用于时空研究和遗传网络的稀疏协方差估计。所有这些问题从它们的应用背景来看都有其显著的特点，但它们都面临着相似的挑战，具有高维性和稀疏性的特点。将通过开发新的统计方法来应对这些新出现的高社会影响的问题，以解决从统计计算、特征选择到降噪等与高维相关的特征和挑战。同时，PI还打算通过渐近分析和模拟研究，为这些问题及其相关方法论提供基础性的理解，推动理论、方法和计算的发展。由于技术创新，如今许多当代科学问题都可以获得大规模和复杂的数据。需要高维统计模型来解决这些科学努力。高维的挑战来自科学和人文的不同领域，从基因组学和卫生科学到经济和金融。在这些领域中，变量选择、特征提取、稀疏挖掘是知识发现的关键。在这项建议中，我们建议从基因组研究、机器学习、健康科学、经济学和金融学等方面发展尖端统计理论和方法来解决这些问题。所提出的技术和结果不仅将帮助研究人员解决其学科中出现的新问题，而且将对统计思维、方法论发展和理论研究产生重大影响。

The challenge of high-dimensionality characterizes many contemporary statistical problems arising frommany frontiers of scientific research and technological development. In high-dimensional statistical research, low-dimensional structures, which entail sparsity under suitable parametrization, are needed to be explored in order to circumvent the issue of noise accumulation with dimensionality. This proposal intends to confront a number of important high-dimensional statistical problems from genomics, machine learning, health studies, economics, and finance. These include various emerging issues from the analysis of microarray data such as normalization, significance analysis, and disease classification; variable selection and feature extraction from high-dimensional statistical learning; sparse classification and clustering from high-dimensional feature spaces; high-dimensional covariance matrix estimation for asset allocation and portfolio management; sparse covariance estimation for spatial and temporal studies and genetic networks. All of these problems have their distinguished characters from the context of their applications, but nevertheless share similar challenges with high dimensionality and admit features of sparsity. These emerging problems of high societal impacts will be confronted via developing new statistical methods to address the features and challenges associated with high-dimensionality, from statistical computation, feature selection, to noise reduction. At the same time, the PI also intends to provide fundamental understanding, via asymptotic analysis and simulation studies, to these problems and their associated methodologies that push theory, methods, and computation forward.Thanks to technological innovation, the availability of large-scale and complex data are widely available nowadays in many contemporary scientific problems. High-dimensional statistical models are required to address these scientific endeavors. The challenges of high-dimensionality arise from diverse fields of sciences and the humanities, ranging from genomics and health sciences to economics and finance. In these fields, variable selection, feature extraction, sparsity explorations are crucial for knowledge discovery. In this proposal, we propose to develop cutting-edge statistical theory and methods to address these problems from genomic studies, machine learning, health science, economics, and finance. The proposed techniques and results will not only help researchers to solve emerging problems in their disciplines, but also have strong impact on statistical thinking, methodological development, and theoretical studies.