A matrix $\Phi \in \mathbb{R}^{Q \times N}$ satisfies the restricted isometry property if $\|\Phi x\|_2^2$ is approximately equal to $\|x\|_2^2$ for all $k$-sparse vectors $x$. We give a construction of RIP matrices with the optimal $Q = O(k \log(N/k))$ rows using $O(k\log(N/k)\log(k))$ bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to $\epsilon$-biased distributions.

翻译：矩阵 $\Phi \in \mathbb{R}^{Q \times N}$ 满足受限等距性质是指：对所有 $k$-稀疏向量 $x$，有 $\|\Phi x\|_2^2$ 近似等于 $\|x\|_2^2$。本文给出一种使用 $O(k\log(N/k)\log(k))$ 比特随机性构造行数最优 $Q = O(k \log(N/k))$ 的RIP矩阵的方法。主要技术要素是将Hanson-Wright不等式推广到$\epsilon$-有偏分布。