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.

翻译：我们研究了植入随机场居里-外斯模型（即pRFCW模型）中隐藏的$k$-团铁磁相关性的检验与恢复问题。pRFCW模型是一种随机效应伊辛模型，其在统计和物理上展现出比标准居里-外斯模型更丰富的相图。通过将参数区域表征为“温度”并将均值视为“外磁场”，我们建立了极小化极大最优检测率和恢复率。结果包含检验问题的$7$个不同相和精确恢复问题的$3$个相。我们的结果还表明，外磁场的随机性导致了可数无穷多个可能的收敛速率，这在固定场模型中并未观察到。作为证明技术的副产品，我们提供了两个新的数学结果：（1）随机场居里-外斯模型（即RFCW模型）在所有温度和任意外场下平均磁化强度的一族尾界；（2）RFCW模型之间信息散度的精确估计。这些结果在建立本文的主要理论结果中发挥了关键作用。此外，我们证明了pRFCW隐团推断问题中涉及的数学结构类似于离散数据的“稀疏PCA型”问题。与长期研究的连续高斯对应模型相比，更丰富的统计相为离散数据的稀疏PCA理论洞察提供了新的视角。