Column normalization of a random measurement matrix

Abstract

In this note we answer a question of G. Lecué, 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 $X \in \mathbb{R} ^d$ with iid, mean-zero, variance $1$ coordinates, that satisfies $\sup _{t \in S^{d-1}} \|\bigl < {X,t} \bigr >\|_{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 $\tilde{\Gamma } :\mathbb{R} ^d \to \mathbb{R} ^m$ 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 $2$.