Improved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems

Given a large data matrix A∈\mathbbR^n\times n, we consider the problem of determining whether its entries are i.i.d. from some known marginal distribution A_ij∼P_0, or instead A contains a principal submatrix A_\sf Q,\sf Q whose entries have marginal distribution A_ij∼P_1≠P_0. As a special case, the hidden (or planted) clique problem is finding a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided |\sf Q|\ge C \log n for a suitable constant C. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when |\sf Q| = o(\sqrtn). Recently, \citemeka2013association proposed a method to establish lower bounds for the hidden clique problem within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree-4 SOS relaxation, and study the construction of \citemeka2013association to prove that SOS fails unless k\ge C\,n^1/3/\log n. An argument presented by \citeBarakLectureNotes implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moment method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erdös-Renyi random graph.