Norms of structured random matrices

Abstract

For m, n ∈ N let X = (X ij ) i≤m,j≤n be a random matrix, A = (a ij ) i≤m,j≤n a real deterministic matrix, and X A = (a ij X ij ) i≤m,j≤n the corresponding structured random matrix. We study the expected operator norm of X A considered as a random operator between ℓ n p and ℓ m q for 1 ≤ p, q ≤ ∞. We prove optimal bounds up to logarithmic terms when the underlying random matrix X has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero ψr (r ∈ (0, 2]) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products A • A and (A • A) T .