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

TL;DR

This paper introduces a simple, threshold-based algorithm for high-dimensional Ising model structure estimation that is efficient for graphs with sparse local separators, requiring minimal samples proportional to the inverse square of the smallest edge potential.

Contribution

It proposes a novel local separation criterion for tractable graph families, enabling efficient structure estimation with proven sample complexity bounds.

Findings

Algorithm achieves structure estimation with $n=\Omega(J_{\min}^{-2}\log p)$ samples.

Introduces a new criterion based on sparse local separators.

Provides necessary and sufficient conditions for accurate estimation.

Abstract

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.