Norms of structured random matrices.

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 p n and ℓ q m 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 ψ r ( 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 .