Bounding Norms of Locally Random Matrices

Recently, several papers proving lower bounds for the performance of the Sum Of Squares Hierarchy on the planted clique problem have been published. A crucial part of all four papers is probabilistically bounding the norms of certain “locally random” matrices. In these matrices, the entries are not completely independent of each other, but rather depend upon a few edges of the input graph. In this paper, we study the norms of these locally random matrices. We start by bounding the norms of simple locally random matrices, whose entries depend on a bipartite graph H and a random graph G; we then generalize this result by bounding the norms of complex locally random matrices, matrices based off of a much more general graph H and a random graph G. For both cases, we prove almost-tight probabilistic bounds on the asymptotic behavior of the norms of these matrices.