Learning Gaussian tree models: analysis of error exponents and extremal structures

The problem of learning tree-structured 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 least-squares 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...

• Type: Journal
• Year: 2010
• Where: TSP
• Authors: Vincent Y. F. Tan, Animashree Anandkumar, Alan S. Willsky
• Keywords: Artificial Intelligence, Exponent, Gaussian Graphical Models, Tree Structure, TSP 2010