Matrices $\Phi\in\R^{n\times p}$ satisfying the Restricted Isometry Property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for $n=\log^{O(1)}p$, the explicit construction of such matrices defied the repeated efforts, and the most known approaches hit the so-called $\sqrt{n}$ sparsity bottleneck. The notable exception is the work by Bourgain et al \cite{bourgain2011explicit} constructing an $n\times p$ RIP matrix with sparsity $s=\Theta(n^{{1\over 2}+\epsilon})$, but in the regime $n=\Omega(p^{1-\delta})$. In this short note we resolve this open question in a sense by showing that an explicit construction of a matrix satisfying the RIP in the regime $n=O(\log^2 p)$ and $s=\Theta(n^{1\over 2})$ implies an explicit construction of a three-colored Ramsey graph on $p$ nodes with clique sizes bounded by $O(\log^2 p)$ -- a question in the extremal combinatorics which has been open for decades.

中文翻译：

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RIP 矩阵的显式构造是 Ramsey-Hard

满足受限等距特性 (RIP) 的矩阵 $\Phi\in\R^{n\times p}$ 是压缩传感方法的重要组成部分。虽然已知随机矩阵即使对于 $n=\log^{O(1)}p$ 也有很高的概率满足 RIP，但这种矩阵的显式构造违背了反复的努力，并且最著名的方法达到了称为 $\sqrt{n}$ 稀疏瓶颈。值得注意的例外是 Bourgain 等人的工作 \cite{bourgain2011explicit} 构造了一个 $n\times p$ RIP 矩阵，稀疏性 $s=\Theta(n^{{1\over 2}+\epsilon})$，但是在 $n=\Omega(p^{1-\delta})$ 状态下。