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 $2$.