Hidden Clique Inference in Random Ising Model II: the planted Sherrington-Kirkpatrick model

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.