1. Statistical Estimation of High Dimensional Covariance Matrices – a sampling from Prof. Hero’s research group Ted Tsiligkaridis SPEECS Friday, Sept. 9, 2011

2. Theme 1 High dimensional statistics Dimensionality reduction Structural graphical models for dynamic spatio- temporal processes Applications: sparsity regularization in inverse problems, functional estimation, covariance matrix estimation, genetic, metabolic regulation networks, dynamics of social networks

3. Theme 2 Distributed, Adaptive and Statistical Signal Processing Computational and Statistical methods in Machine Learning Applications: Anomaly detection, localization, tracking, imaging, clustering, semi-supervised classification, pattern matching, multimodality image registration, database indexing and retrieval

4. High dimensional sparse covariance estimation with special structural constraints Consider the simple setting of n i.i.d. zero-mean MVN data of dimension d. How to estimate covariance matrix? Naïve approach: form Sample Covariance Matrix But for small sample regime (n<d), this is singular! Also, poor performance for small-sample regime.

5.  What to do? If precision matrix is sparse, consistent estimators of true precision matrix exist (penalized maximum likelihood), even if n<d. High dimensional sparse covariance estimation with special structural constraints

6. Extend this framework to covariance matrices with special structure. Contributions: develop estimators that exploit structure and sparsity, performance analysis in different regimes & simulations Applications in wireless communications, modeling social networks and gene networks High dimensional sparse covariance estimation with special structural constraints

8. Thank you, and welcome to Michigan!