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.

翻译：称矩阵$A$具有$\ell_p$-限制等距性质（$\ell_p$-RIP），若对所有稀疏度不超过$k$的向量$x$，$\|{Ax}\|_p$大致与$\|{x}\|_p$成比例。我们研究了秩与$n$成比例且$k = \Theta(n)$的$m \times n$矩阵的这一性质。在该参数范围内，$\ell_p$-RIP矩阵与欧几里得截面紧密相关，并且是局部可测试码的测试矩阵的“实模拟”。已知随机稠密$m\times n$矩阵（例如具有独立同分布$\pm 1$项的矩阵）以高概率满足$\ell_2$-RIP且$k \approx m/\log n$；而稀疏随机矩阵在$p \in [1,2)$且$k, m = \Theta(n)$时满足$\ell_p$-RIP。然而，当$m = \Theta(n)$时，已知稀疏随机矩阵以高概率不满足$\ell_2$-RIP。在此背景下，我们证明稀疏矩阵在我们的参数范围内不可能满足$\ell_2$-RIP。另一方面，对于$p \neq 2$，我们证明每个$\ell_p$-RIP矩阵必须是稀疏的。因此，稀疏性与$\ell_2$-RIP不相容，但对于$p \neq 2$的$\ell_p$-RIP却是必要的。在适当解释下，我们关于$\ell_2$-RIP的否定结果给出了某个连续的“$c^3$-LTC”模拟的一个不可能性结果：即近期突破性成果中构建的具有恒定速率、恒定距离和恒定局部性的局部可测试码。