High-Temperature Structure Detection in Ferromagnets
This addresses a fundamental statistical inference problem for researchers in statistical physics and machine learning, though it is incremental in building on existing graph testing frameworks.
The paper tackles the problem of detecting specific graph structures in high-temperature ferromagnetic Ising models, establishing matching minimax bounds that show testability is determined by graph arboricity, and proving computational hardness results under a conjecture about sparse PCA.
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.