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High-dimensional Ising model selection using $$\ell _{1}$$-regularized logistic regression. (English) Zbl 1189.62115
Summary: We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on $$\ell _{1}$$-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an $$\ell_{1}$$-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes $$p$$ and maximum neighborhood size $$d$$ are allowed to grow as a function of the number of observations $$n$$. Our main results provide sufficient conditions on the triple $$(n, p, d)$$ and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously.
With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes $$n=\Omega (d^{3} \log p)$$ with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of $$n=\Omega (d^{2} \log p)$$ suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.

##### MSC:
 62J12 Generalized linear models (logistic models) 62F12 Asymptotic properties of parametric estimators 05C90 Applications of graph theory 68T99 Artificial intelligence 62M40 Random fields; image analysis