Column normalization of a random measurement matrix

In this note we answer a question of G. Lecu\'{e}, by showing that column normalization of a random matrix with iid entries need not lead to good sparse recovery properties, even if the generating random variable has a reasonable moment growth. Specifically, for every $2 \leq p \leq c_1\log d$ we construct a random vector $X \in R^d$ with iid, mean-zero, variance $1$ coordinates, that satisfies $\sup_{t \in S^{d-1}} \|<X,t>\|_{L_q} \leq c_2\sqrt{q}$ for every $2\leq q \leq p$. We show that if $m \leq c_3\sqrt{p}d^{1/p}$ and $\tilde{\Gamma}:R^d \to R^m$ is the column-normalized matrix generated by $m$ independent copies of $X$, then with probability at least $1-2\exp(-c_4m)$, $\tilde{\Gamma}$ does not satisfy the exact reconstruction property of order $2$.