Improved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems
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Given a large data matrix A∈R^n× n, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution A_ij∼ P_0, or instead A contains a principal submatrix A_ Q, Q whose entries have marginal distribution A_ij∼ P_1≠ P_0. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided | Q|> C n for a suitable constant C. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when | Q| = o(√(n)). Recently Meka and Wigderson meka2013association, proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree-4 SOS relaxation, and study the construction of meka2013association to prove that SOS fails unless k> C n^1/3/ n. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments 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.
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