How do the structure and the parameters of Gaussian tree models affect structure learning?
Author(s)Tan, Vincent Yan Fu; Anandkumar, Animashree; Willsky, Alan S.
DownloadTan-2009-How do the structure and the parameters of Gaussian tree models affect structure learning.pdf (264.5Kb)
MetadataShow full item record
The problem of learning tree-structured Gaussian graphical models from 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 universally, the extremal tree structures which maximize and minimize the error exponent are the star and the Markov chain for any fixed set of correlation coefficients on the edges of the tree. In other words, the star and the chain graphs represent the hardest and the easiest structures to learn in the class of tree-structured 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 maximum-likelihood estimator in the asymptotic regime.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Laboratory for Information and Decision Systems; Massachusetts Institute of Technology. Stochastic Systems Group
47th Annual Allerton Conference on Communication, Control, and Computing, 2009. Allerton 2009
Institute of Electrical and Electronics Engineers
Tan, Vincent Y. F., Animashree Anandkumar, and Alan S. Willsky. “How Do the Structure and the Parameters of Gaussian Tree Models Affect Structure Learning?” Communication, Control, and Computing, 2009. Allerton 2009. 47th Annual Allerton Conference On. 2009. 684-691. © 2010 IEEE.
Final published version
INSPEC Accession Number: 11135184