HIGH-DIMENSIONAL ISING MODEL SELECTION USING \(\ell_1\)-REGULARIZED LOGISTIC REGRESSION

HIGH-DIMENSIONAL ISING MODEL SELECTION USING \(\ell_1\)-REGULARIZED LOGISTIC REGRESSION

2010, Vol. 38, No. 3, 1287–1319 | BY PRADEEP RAVIKUMAR\(^{1,2,3}\), MARTIN J. WAINWRIGHT\(^{3}\) AND JOHN D. LAFFERTY\(^{1}\)
The paper presents a method for high-dimensional Ising model selection using $\ell_1$-regularized logistic regression. The method estimates the neighborhood of each node in a binary Ising Markov random field by performing logistic regression with an $\ell_1$-constraint. The analysis is conducted under high-dimensional scaling, where both the number of nodes $p$ and maximum neighborhood size $d$ grow with the number of observations $n$. The main result shows that consistent neighborhood selection is possible with sample sizes $n = \Omega(d^3 \log p)$, and a reduced sample size $n = \Omega(d^2 \log p)$ suffices when conditions are imposed directly on the sample matrices. The method is computationally efficient, avoiding the need for computing the normalization constant or combinatorial search through graph structures. It is applicable to binary Ising models and can be generalized to other discrete Markov random fields. The paper also discusses the theoretical guarantees of the method, including conditions for sparsistency and the probability of correct model selection. The analysis involves high-dimensional statistical methods, including concentration inequalities and properties of the Fisher information matrix. The method is shown to be effective in recovering the true graph structure under appropriate scaling conditions.The paper presents a method for high-dimensional Ising model selection using $\ell_1$-regularized logistic regression. The method estimates the neighborhood of each node in a binary Ising Markov random field by performing logistic regression with an $\ell_1$-constraint. The analysis is conducted under high-dimensional scaling, where both the number of nodes $p$ and maximum neighborhood size $d$ grow with the number of observations $n$. The main result shows that consistent neighborhood selection is possible with sample sizes $n = \Omega(d^3 \log p)$, and a reduced sample size $n = \Omega(d^2 \log p)$ suffices when conditions are imposed directly on the sample matrices. The method is computationally efficient, avoiding the need for computing the normalization constant or combinatorial search through graph structures. It is applicable to binary Ising models and can be generalized to other discrete Markov random fields. The paper also discusses the theoretical guarantees of the method, including conditions for sparsistency and the probability of correct model selection. The analysis involves high-dimensional statistical methods, including concentration inequalities and properties of the Fisher information matrix. The method is shown to be effective in recovering the true graph structure under appropriate scaling conditions.
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