Operator ℓp → ℓq norms of random matrices with iid entries

Abstract

We prove that for every p,q∈[1,∞] and every random matrix X=(Xi,j)i≤m,j≤n with iid centered entries satisfying the α-regularity assumption ‖Xi,j‖2ρ≤α‖Xi,j‖ρ for every ρ≥1, the expectation of the operator norm of X from ℓpn to ℓqm is comparable, up to a constant depending only on α, tom1/qsupt∈Bpn⁡‖∑j=1ntjX1,j‖q∧Logm+n1/p⁎sups∈Bq⁎m⁡‖∑i=1msiXi,1‖p⁎∧Logn. We give more explicit formulas, expressed as exact functions of p, q, m, and n, for the two-sided bounds of the operator norms in the case when the entries Xi,j are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range 1≤q≤2≤p we provide two-sided bounds under the weaker regularity assumption (EX1,14)1/4≤α(EX1,12)1/2.