Oracle Inequalities in Sparse Regression and Low Rank Matrix Estimation

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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.