Learning Gaussian Tree Models: Analysis of Error Exponents and Extremal Structures
The problem of learning treestructured Gaussian graphical models from independent and identically distributed (i.i.d.) samples is considered. The influence of the tree structure and the parameters of the Gaussian distribution on the learning rate as the number of samples increases is discussed. Specifically, the error exponent corresponding to the event that the estimated tree structure differs from the actual unknown tree structure of the distribution is analyzed. Finding the error exponent reduces to a leastsquares problem in the very noisy learning regime. In this regime, it is shown that the extremal tree structure that minimizes the error exponent is the star for any fixed set of correlation coefficients on the edges of the tree. If the magnitudes of all the correlation coefficients are less than 0.63, it is also shown that the tree structure that maximizes the error exponent is the Markov chain. In other words, the star and the chain graphs represent the hardest and the easiest structures to learn in the class of treestructured Gaussian graphical models. This result can also be intuitively explained by correlation decay: pairs of nodes which are far apart, in terms of graph distance, are unlikely to be mistaken as edges by the maximumlikelihood estimator in the asymptotic regime.
Year of publication: 
200809


Authors:  Tan, Vincent Y. F. ; Anandkumar, Animashree ; Willsky, Alan S. 
Publisher: 
Institute of Electrical and Electronics Engineers 
Saved in favorites
Similar items by person

How do the structure and the parameters of Gaussian tree models affect structure learning?
Tan, Vincent Yan Fu, (2009)

Gaussian Multiresolution Models: Exploiting Sparse Markov and Covariance Structure
Choi, Myung Jin, (2009)

Nonparametric Belief Propagation and Facial Appearance Estimation
Sudderth, Erik B., (2002)
 More ...