The main focus of the lectures is to discuss new results and current research problems in high dimensional statistical inference, which is one of the most active research areas in statistics at the moment. These and other related problems have also attracted much recent interest in other fields including applied mathematics and electrical engineering.
To provide a strong background and foundation for the main topics, we shall begin with discussions on important results in nonparametric function estimation in the framework of the infinite dimensional Gaussian sequence model. Minimaxity, adaptive minimaxity, and oracle inequalities are covered in the context of the sequence model. In particular, Pinsker's results on linear minimaxity for estimation over an ellipsoid and the wavelet thresholding theory developed by Donoho and Johnstone will be discussed. We will then focus on current research problems in high dimensional inference including compressed sensing (large p, small n linear regression), detection of sparse signals, and estimation of large covariance matrices. We specifically cover in detail the constrained l1 minimization methods and present a unified and elementary analysis on sparse signal recovery in three settings: noiseless, bounded noise and Gaussian noise. In addition, new results on optimal estimation of large covariance matrices are presented. The analysis of the matrix estimation problems reveals new features that are quite different from those in the more conventional function/sequence estimation problems.