Abstract
We prove estimates for $\mathbb{E} \| X: \ell _{p'}^{n} \to \ell _{q}^{m}\|$ for $p,q\ge 2$ and any random matrix $X$ having the entries of the form $a_{ij}Y_{ij}$, where $Y=(Y_{ij})_{1\le i\le m, 1\le j\le n}$ has i.i.d. isotropic log-concave rows and $p'$ denotes the Holder conjugate of $p$. This generalises a result of Guedon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide an analogous bound for $m\times n$ random matrices, whose entries form an unconditional vector in $\mathbb{R} ^{mn}$. We also prove bounds for norms of matrices whose entries are certain Gaussian mixtures.
Highlights
Introduction and main resultsBy }A}p,q we denote the operator norm of the matrix A from p to q
Note that in (1.3) the logarithmic term appears in front of the norm of rows, but not in front of the norm of columns, so our bound is not symmetric
This is not so strange, since the assumptions of the theorem are non-symmetric: we assume that the rows are weighted i.i.d. random vectors, but no independence between the columns is required
Summary
By }A}p,q we denote the operator norm of the matrix A from p to q. Note that in (1.3) the logarithmic term appears in front of the norm of rows, but not in front of the norm of columns, so our bound is not symmetric This is not so strange, since the assumptions of the theorem are non-symmetric: we assume that the rows are weighted i.i.d. random vectors, but no independence between the columns is required. If the rows and columns of a random matrix Y are isotropic and log-concave (we do not require independence), and p, q ě 1, max nÿ1{p |Aij Yij |p max mÿ1{q |Aij Yij |q. This means that inequality (1.4) may be reversed up to a logarithmic factor and constants depending only on p and q in the log-concave setting. For p P r1, 8s we denote by p1 the Hölder conjugate of p, i.e
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