Norms of structured random matrices

Abstract

AbstractFor $$m,n\in \mathbb {N}$$ m , n ∈ N , let $$X=(X_{ij})_{i\le m,j\le n}$$ X = ( X ij ) i ≤ m , j ≤ n be a random matrix, $$A=(a_{ij})_{i\le m,j\le n}$$ A = ( a ij ) i ≤ m , j ≤ n a real deterministic matrix, and $$X_A=(a_{ij}X_{ij})_{i\le m,j\le n}$$ X A = ( a ij X ij ) i ≤ m , j ≤ n the corresponding structured random matrix. We study the expected operator norm of $$X_A$$ X A considered as a random operator between $$\ell _p^n$$ ℓ p n and $$\ell _q^m$$ ℓ q m for $$1\le p,q \le \infty $$ 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 $$\psi _r$$ ψ r ($$r\in (0,2]$$ 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 $$A\circ A$$ A ∘ A and $$(A\circ A)^T$$ ( A ∘ A ) T .