Statistical Estimation of High
Dimensional Covariance Matrices
– a sampling from Prof. Hero’s research group
Ted Tsiligkaridis
SPEECS
Friday, Sept. 9, 2011
Theme 1
• High dimensional statistics
• Dimensionality reduction
• Structural graphical models for dynamic spatiotemporal processes
Applications:
sparsity regularization in inverse problems, functional
estimation, covariance matrix estimation, genetic,
metabolic regulation networks, dynamics of social
networks
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
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.
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.
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
High dimensional sparse covariance estimation
with special structural constraints
Thank you, and welcome to
Michigan!