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

by   Vincent Y. F. Tan, et al.

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.


page 1

page 2

page 3

page 4


Active-LATHE: An Active Learning Algorithm for Boosting the Error Exponent for Learning Homogeneous Ising Trees

The Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) has been a mai...

SGA: A Robust Algorithm for Partial Recovery of Tree-Structured Graphical Models with Noisy Samples

We consider learning Ising tree models when the observations from the no...

Learning High-Dimensional Mixtures of Graphical Models

We consider unsupervised estimation of mixtures of discrete graphical mo...

New Optimisation Methods for Machine Learning

A thesis submitted for the degree of Doctor of Philosophy of The Austral...

Structure learning of antiferromagnetic Ising models

In this paper we investigate the computational complexity of learning th...

Learning of Tree-Structured Gaussian Graphical Models on Distributed Data under Communication Constraints

In this paper, learning of tree-structured Gaussian graphical models fro...

Exact Asymptotics for Learning Tree-Structured Graphical Models with Side Information: Noiseless and Noisy Samples

Given side information that an Ising tree-structured graphical model is ...

Please sign up or login with your details

Forgot password? Click here to reset