Estimates of norms of log-concave random matrices with dependent entries

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. This generalises the result of Gu\'edon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide the analogue bound for $m\times n$ random matrices, which entries form an unconditional vector in $\mathbb{R}^{mn}$. We also prove bounds for norms of matrices which entries are certain Gaussian mixtures.