Hidden Clique Inference in Random Ising Model I: the planted random field Curie-Weiss model

We study the problem of testing and recovering the hidden $k$-clique Ferromagnetic correlation in the planted Random Field Curie-Weiss model (a.k.a. the pRFCW model). The pRFCW model is a random effect Ising model that exhibits richer phase diagrams both statistically and physically than the standard Curie-Weiss model. Using an alternative characterization of parameter regimes as 'temperatures' and the mean values as 'outer magnetic fields,' we establish the minimax optimal detection rates and recovery rates. The results consist of $7$ distinctive phases for testing and $3$ phases for exact recovery. Our results also imply that the randomness of the outer magnetic field contributes to countable possible convergence rates, which are not observed in the fixed field model. As a byproduct of the proof techniques, we provide two new mathematical results: (1) A family of tail bounds for the average magnetization of the Random Field Curie-Weiss model (a.k.a. the RFCW model) across all temperatures and arbitrary outer fields. (2) A sharp estimate of the information divergence between RFCW models. These play pivotal roles in establishing the major theoretical results in this paper. Additionally, we show that the mathematical structure involved in the pRFCW hidden clique inference problem resembles a 'sparse PCA-like' problem for discrete data. The richer statistical phases than the long-studied Gaussian counterpart shed new light on the theoretical insight of sparse PCA for discrete data.