The Annals of Statistics

High-dimensional structure estimation in Ising models: Local separation criterion

Animashree Anandkumar, Vincent Y. F. Tan, Furong Huang, and Alan S. Willsky

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We consider the problem of high-dimensional Ising (graphical) model selection. We propose a simple algorithm for structure estimation based on the thresholding of the empirical conditional variation distances. We introduce a novel criterion for tractable graph families, where this method is efficient, based on the presence of sparse local separators between node pairs in the underlying graph. For such graphs, the proposed algorithm has a sample complexity of $n=\Omega(J_{\min}^{-2}\log p)$, where $p$ is the number of variables, and $J_{\min}$ is the minimum (absolute) edge potential in the model. We also establish nonasymptotic necessary and sufficient conditions for structure estimation.

Article information

Ann. Statist., Volume 40, Number 3 (2012), 1346-1375.

First available in Project Euclid: 10 August 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 05C80: Random graphs [See also 60B20]

Ising models graphical model selection local-separation property


Anandkumar, Animashree; Tan, Vincent Y. F.; Huang, Furong; Willsky, Alan S. High-dimensional structure estimation in Ising models: Local separation criterion. Ann. Statist. 40 (2012), no. 3, 1346--1375. doi:10.1214/12-AOS1009.

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