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：我们研究在具有n个自旋的植入式谢林顿-柯克帕特里克（Sherrington-Kirkpatrick）模型（即一种自旋玻璃模型）中，对k团铁磁均值漂移进行检验与恢复的问题。植入式SK模型（一种由不可数无穷多个伊辛模型构成的典型混合体）使我们能够研究在模型误设条件下相依随机变量相关性分析的基本极限。本文的三大主要贡献如下：（i）通过为多个不同参数区间提供极小极大最优速率，我们识别了检验问题的相图；此外，我们还给出了高温/临界温区和低温温区精确恢复的极小极大最优速率。（ii）我们证明了一个普适性结果，表明所有得到的速率在非高斯耦合条件下仍然成立。（iii）为实现上述主要结果，我们还针对植入式SK模型局部与全局相的平均统计量，建立了一系列新的集中不等式和中心极限定理。这些技术性结果为植入式自旋玻璃模型提供了新的洞见。植入式SK模型还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，后者将背景单位精度矩阵替换为维格纳随机矩阵。