Ted Tsiligkaridis SPEECS Friday, Sept. 9, 2011

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

6. High dimensional sparse covariance estimation with special structural constraints • 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

7. High dimensional sparse covariance estimation with special structural constraints

8. Thank you, and welcome to Michigan!