High-Temperature Structure Detection in Ferromagnets

TL;DR

This paper investigates the problem of detecting specific structures in high-temperature ferromagnetic Ising models, establishing fundamental limits and computational hardness results based on graph properties and moment inequalities.

Contribution

It provides the first minimax bounds for structure detection in high-temperature ferromagnetic Ising models and links testability to graph arboricity, also exploring computational hardness under a conjecture.

Findings

Matching upper and lower bounds for detection are established.

Testability depends on graph arboricity.

Computational hardness results are derived under a conjecture.

Abstract

This paper studies structure detection problems in high temperature ferromagnetic (positive interaction only) Ising models. The goal is to distinguish whether the underlying graph is empty, i.e., the model consists of independent Rademacher variables, versus the alternative that the underlying graph contains a subgraph of a certain structure. We give matching upper and lower minimax bounds under which testing this problem is possible/impossible respectively. Our results reveal that a key quantity called graph arboricity drives the testability of the problem. On the computational front, under a conjecture of the computational hardness of sparse principal component analysis, we prove that, unless the signal is strong enough, there are no polynomial time tests which are capable of testing this problem. In order to prove this result we exhibit a way to give sharp inequalities for the even moments of sums of i.i.d. Rademacher random variables which may be of independent interest.