Operator ${\ell }_p\to {\ell }_q$ norms of random matrices with iid entries

Abstract

We prove that for every $p,q\in [1,\infty ]$ and every random matrix $X=(X_{i,j}{)}_{i\le m,j\le n}$ with iid centered entries satisfying the 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 $\alpha$, to \[ m^{1/q}\sup_{t\in B_p^n}\Bigl\|\sum_{j=1}^nt_jX_{1,j}\Bigr\|_{ q\wedge \operatorname{Log} m} +n^{1/p^*}\sup_{s\in B_{q^*}^m}\Bigl\|\sum_{i=1}^{m} s_iX_{i,1}\Bigr\|_{ p^*\wedge \operatorname{Log} n}. \] We give more explicit formulas, expressed as exact functions of $p$, $q$, $m$, and $n$, for the asymptotic 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 a weaker regularity assumption $(\mathbb{E}{X}_{1,1}^4{)}^{1/4}\le \alpha (\mathbb{E}{X}_{1,1}^2{)}^{1/2}$.

28 pages

References in corpus (1)

• Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices