Operator ℓp → ℓq norms of random matrices with iid entries

Abstract

We prove that for every $p,q\in [1,\infty ]$ and every random matrix $X={\left(X_{i,j}\right)}_{i\le m,j\le n}$ with iid centered entries satisfying the α-regularity assumption ${\Vert X_{i,j}\Vert }_{2\rho }\le \alpha {\Vert X_{i,j}\Vert }_{\rho }$ for every $\rho \ge 1$, the expectation of the operator norm of X from ${\ell }_p^n$ to ${\ell }_q^m$ is comparable, up to a constant depending only on α, to$m^{1/q}\underset{t\in B_p^n}{\sup }{\Vert \sum _{j=1}^n{t}_j{X}_{1,j}\Vert }_{q\wedge \mathrm{Log}m}+n^{1/p^{⁎}}\underset{s\in B_{q^{⁎}}^m}{\sup }{\Vert \sum _{i=1}^m{s}_i{X}_{i,1}\Vert }_{p^{⁎}\wedge \mathrm{Log}n}.$ 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 $X_{i,j}$ are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range $1\le q\le 2\le p$ we provide two-sided bounds under the weaker regularity assumption ${\left(E{X}_{1,1}^4\right)}^{1/4}\le \alpha {\left(E{X}_{1,1}^2\right)}^{1/2}$.