On Model Selection Consistency of Lasso for High-Dimensional Ising Models on Tree-like Graphs

We consider the problem of high-dimensional Ising model selection using neighborhood-based least absolute shrinkage and selection operator (Lasso). It is rigorously proved that under some mild coherence conditions on the population covariance matrix of the Ising model, consistent model selection can be achieved with sample sizes $n=\Omega{(d^3\log{p})}$ for any tree-like graph in the paramagnetic phase, where $p$ is the number of variables and $d$ is the maximum node degree. When the same conditions are imposed directly on the sample covariance matrices, it is shown that a reduced sample size $n=\Omega{(d^2\log{p})}$ suffices. The obtained sufficient conditions for consistent model selection with Lasso are the same in the scaling of the sample complexity as that of $\ell_1$-regularized logistic regression. Given the popularity and efficiency of Lasso, our rigorous analysis provides a theoretical backing for its practical use in Ising model selection.