Learning Gaussian Tree Models: Analysis of Error Exponents and Extremal Structures

TL;DR

This paper analyzes the error rates in learning Gaussian tree models from data, revealing that star and chain structures are respectively the hardest and easiest to learn, depending on correlation strengths and graph topology.

Contribution

It characterizes the error exponents for structure learning in Gaussian trees and identifies extremal structures that minimize or maximize learning difficulty.

Findings

Star structure minimizes the error exponent, making it hardest to learn.

Chain structure maximizes the error exponent, making it easiest to learn.

Correlation decay explains the difficulty in distinguishing distant nodes.

Abstract

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 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.