Necessary moment conditions for exact reconstruction via basis pursuit

Let $X=(x_1,...,x_n)$ be a random vector that satisfies a weak small ball property and whose coordinates $x_i$ satisfy that $\|x_i\|_{L_p} \lesssim \sqrt{p} \|x_i\|_{L_2}$ for $p \sim \log n$. In \cite{LM_compressed}, it was shown that $N$ independent copies of $X$ can be used as measurement vectors in Compressed Sensing (using the basis pursuit algorithm) to reconstruct any $d$-sparse vector with the optimal number of measurements $N\gtrsim d \log\big(e n/d\big)$. In this note we show that the result is almost optimal. We construct a random vector $X$ with iid, mean-zero, variance one coordinates that satisfies the same weak small ball property and whose coordinates satisfy that $\|x_i\|_{L_p} \lesssim \sqrt{p} \|x_i\|_{L_2}$ for $p \sim (\log n)/(\log N)$, but the basis pursuit algorithm fails to recover even $1$-sparse vectors. The construction shows that `spiky' measurement vectors may lead to a poor performance by the basis pursuit algorithm, but on the other hand may still perform in an optimal way if one chooses a different reconstruction algorithm (like $\ell_0$-minimization). This exhibits the fact that the convex relaxation of $\ell_0$-minimization comes at a significant cost when using `spiky' measurement vectors.