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April 2013 Learning loopy graphical models with latent variables: Efficient methods and guarantees
Animashree Anandkumar, Ragupathyraj Valluvan
Ann. Statist. 41(2): 401-435 (April 2013). DOI: 10.1214/12-AOS1070

Abstract

The problem of structure estimation in graphical models with latent variables is considered. We characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees. We consider models where the underlying Markov graph is locally tree-like, and the model is in the regime of correlation decay. For the special case of the Ising model, the number of samples $n$ required for structural consistency of our method scales as $n=\Omega(\theta_{\min}^{-\delta\eta(\eta+1)-2}\log p)$, where $p$ is the number of variables, $\theta_{\min}$ is the minimum edge potential, $\delta$ is the depth (i.e., distance from a hidden node to the nearest observed nodes), and $\eta$ is a parameter which depends on the bounds on node and edge potentials in the Ising model. Necessary conditions for structural consistency under any algorithm are derived and our method nearly matches the lower bound on sample requirements. Further, the proposed method is practical to implement and provides flexibility to control the number of latent variables and the cycle lengths in the output graph.

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Animashree Anandkumar. Ragupathyraj Valluvan. "Learning loopy graphical models with latent variables: Efficient methods and guarantees." Ann. Statist. 41 (2) 401 - 435, April 2013. https://doi.org/10.1214/12-AOS1070

Information

Published: April 2013
First available in Project Euclid: 16 April 2013

zbMATH: 1267.62070
MathSciNet: MR3099108
Digital Object Identifier: 10.1214/12-AOS1070

Subjects:
Primary: 62H12
Secondary: 05C12

Keywords: graphical model selection , latent variables , quartet methods

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • April 2013
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