Oracle Inequalities in Sparse Regression and Low Rank Matrix Estimation

本课题的主要研究目标与近年来备受关注的高维统计领域相关，如稀疏回归或低秩矩阵估计问题。研究者打算开发新的方法和新的应用，并扩大惩罚经验风险最小化，经验过程和指数权重估计器的应用范围。研究者特别研究了噪声矩阵补全问题中的极大极小率问题，并打算将矩阵补全问题中的成功技术应用于协方差估计问题，并在低秩假设下确定该问题的极大极小率问题。本研究项目的理论成果也有望有更广泛的应用。新的研究成果可以应用于许多领域：计量经济学、市场营销、数据挖掘、量子物理学、宇宙学、基因组学、断层扫描、气候学和许多其他需要有效工具来探索高维数据集的领域。特别是，在所有这些应用程序中，一个至关重要的问题是在大量潜在候选变量中确定一组活动变量。在基因组学中，微阵列芯片包含数千个基因的表达，其目标是在整个被测试基因池中找到负责合成特定分子的少数基因。通过本课题所研究的技术，可以有效地解决这一难题。

The main research objectives of this proposal are related to the field of high-dimensional statistics such as sparse regression or low rank matrix estimation problems which have recently attracted a lot of attention. The investigator intends to develop new methodologies and novel applications and extend the scope of applications for penalized empirical risk minimization, empirical processes and exponential weights estimators. The investigator studies in particular the minimax rates in the noisy matrix completion problem and intends to adapt successful techniques from matrix completion problem to the covariance estimation problem and determine the minimax rate for this problem under the low rank assumption.The theoretical results developed in this research project are expected to have broader applications as well. The new research results can be applied in many fields: econometrics, marketing, data mining, quantum physics, cosmology, genomic, tomography, climatology and many other fields that require efficient tools for exploring high-dimensional data sets. In particular, a question of crucial interest in all these applications is to determine the set of active variables among a huge set of potential candidates. In genomic, micro-array chip contain the expression of thousands of genes and the goal is to find the few genes responsible for the synthesis of a particular molecule among the entire pool of tested genes. This difficult problem can be tackled efficiently through the techniques studied in this research project.