Quantitative estimates of the spectral norm of random matrices with independent columns

This paper investigates the nonasymptotic properties of the spectral norm of some random matrices with independent columns. In particular, we consider an $m\times n$ random matrix $BA$, where $A$ is an $N\times n$ random matrix with independent mean-zero subexponential entries, and $B$ is an $m\times N$ deterministic matrix. We prove that the $L_p$ norm of the spectral norm of $BA$ is upper bounded by $(\sqrt{m}+\sqrt{n})p$. It is remarkable that this result is independent of the dimension $N$.