Efficiently learning Ising models on arbitrary graphs

TL;DR

This paper introduces a simple greedy algorithm that efficiently reconstructs the structure of Ising models on arbitrary bounded-degree graphs in quadratic time, without restrictive assumptions on model parameters.

Contribution

It presents a novel structural property of Ising models and a greedy method that significantly reduces computational complexity for structure learning.

Findings

Reconstructs Ising model graphs in $p^2$ time

Works at low temperatures and with non-uniform models

Identifies a key structural property of high mutual information neighbors

Abstract

We consider the problem of reconstructing the graph underlying an Ising model from i.i.d. samples. Over the last fifteen years this problem has been of significant interest in the statistics, machine learning, and statistical physics communities, and much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models. Nevertheless, for learning Ising models on general graphs with $p$ nodes of degree at most $d$, it is not known whether or not it is possible to improve upon the $p^{d}$ computation needed to exhaustively search over all possible neighborhoods for each node. In this paper we show that a simple greedy procedure allows to learn the structure of an Ising model on an arbitrary bounded-degree graph in time on the order of $p^2$. We make no assumptions on the parameters except what is necessary for identifiability of the model, and in particular the results hold at low-temperatures as well as for highly non-uniform models. The proof rests on a new structural property of Ising models: we show that for any node there exists at least one neighbor with which it has a high mutual information. This structural property may be of independent interest.