We study the problem of testing and recovering $k$-clique Ferromagnetic mean shift in the planted Sherrington-Kirkpatrick model (i.e., a type of spin glass model) with $n$ spins. The planted SK model -- a stylized mixture of an uncountable number of Ising models -- allows us to study the fundamental limits of correlation analysis for dependent random variables under misspecification. Our paper makes three major contributions: (i) We identify the phase diagrams of the testing problem by providing minimax optimal rates for multiple different parameter regimes. We also provide minimax optimal rates for exact recovery in the high/critical and low temperature regimes. (ii) We prove a universality result implying that all the obtained rates still hold with non-Gaussian couplings. (iii) To achieve the major results, we also establish a family of novel concentration bounds and central limiting theorems for the averaging statistics in the local and global phases of the planted SK model. These technical results shed new insights into the planted spin glass models. The pSK model also exhibits close connections with a binary variant of the single spike Gaussian sparse principle component analysis model by replacing the background identity precision matrix with a Wigner random matrix.

翻译：我们研究了植入型Sherrington-Kirkpatrick模型（一种自旋玻璃模型）中$k$团簇铁磁均值偏移的检验与恢复问题，该模型包含$n$个自旋。植入型SK模型——一种不可数伊辛模型的典型混合——使我们能够研究在模型误设条件下相依随机变量相关分析的基本极限。本文做出三大贡献：（i）通过提供多个不同参数区域的最小最大最优检验速率，我们识别了检验问题的相图。同时，我们给出了高/临界温和低温区域内精确恢复的最小最大最优速率。（ii）我们证明了一个普适性结果，表明所有获得的速率在非高斯耦合条件下仍成立。（iii）为达成主要结果，我们还建立了一系列新颖的集中不等式以及关于植入型SK模型局部与全局阶段平均统计量的中心极限定理。这些技术性结果为植入型自旋玻璃模型提供了新见解。此外，通过将背景单位精度矩阵替换为Wigner随机矩阵，植入型SK模型与单尖峰高斯稀疏主成分分析模型的二值变体存在紧密联系。