Sparsity and $\ell_p$-Restricted Isometry

A matrix $A$ is said to have the $\ell_p$-Restricted Isometry Property ($\ell_p$-RIP) if for all vectors $x$ of up to some sparsity $k$, $\|{Ax}\|_p$ is roughly proportional to $\|{x}\|_p$. We study this property for $m \times n$ matrices of rank proportional to $n$ and $k = \Theta(n)$. In this parameter regime, $\ell_p$-RIP matrices are closely connected to Euclidean sections, and are "real analogs" of testing matrices for locally testable codes. It is known that with high probability, random dense $m\times n$ matrices (e.g., with i.i.d. $\pm 1$ entries) are $\ell_2$-RIP with $k \approx m/\log n$, and sparse random matrices are $\ell_p$-RIP for $p \in [1,2)$ when $k, m = \Theta(n)$. However, when $m = \Theta(n)$, sparse random matrices are known to not be $\ell_2$-RIP with high probability. Against this backdrop, we show that sparse matrices cannot be $\ell_2$-RIP in our parameter regime. On the other hand, for $p \neq 2$, we show that every $\ell_p$-RIP matrix must be sparse. Thus, sparsity is incompatible with $\ell_2$-RIP, but necessary for $\ell_p$-RIP for $p \neq 2$. Under a suitable interpretation, our negative result about $\ell_2$-RIP gives an impossibility result for a certain continuous analog of "$c^3$-LTCs": locally testable codes of constant rate, constant distance and constant locality that were constructed in recent breakthroughs.