In this paper, we study the problem of compressed sensing using binary measurement matrices and $\ell _1$-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and show that the associated sufficient conditions for robust sparse recovery obtained using the RNSP are better by a factor of $(3 \sqrt{3})/2 \approx 2.6$ compared to the sufficient conditions obtained using the restricted isometry property (RIP). Next we derive universal lower bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP and show that bipartite graphs of girth six are optimal. Then we display two classes of binary matrices, namely parity check matrices of array codes and Euler squares, which have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle, randomly generated Gaussian measurement matrices are “order-optimal.” So we compare the phase transition behavior of the basis pursuit formulation using binary array codes and Gaussian matrices and show that (i) there is essentially no difference between the phase transition boundaries in the two cases and (ii) the CPU time of basis pursuit with binary matrices is hundreds of times faster than with Gaussian matrices and the storage requirements are less. Therefore it is suggested that binary matrices are a viable alternative to Gaussian matrices for compressed sensing using basis pursuit.

中文翻译：

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使用接近最优尺寸的二元矩阵进行压缩感知

在本文中，我们研究了使用二进制测量矩阵和 $\ell_1$-norm 最小化（基数追踪）作为恢复算法的压缩感知问题。我们推导出新的测量数量的上限和下限，以使用二进制矩阵实现稳健的稀疏恢复。我们为列正则二元矩阵建立了充分条件以满足鲁棒零空间属性 (RNSP)，并表明使用 RNSP 获得的鲁棒稀疏恢复的相关充分条件比 $(3 \sqrt{3} )/2 \approx 2.6$ 与使用受限等距属性 (RIP) 获得的充分条件相比。接下来，我们推导出任何二元矩阵需要具有的测量数量的通用下限，以满足基于 RNSP 的较弱充分条件，并表明周长 6 的二部图是最佳的。然后我们显示两类二进制矩阵，即数组代码的奇偶校验矩阵和欧拉平方，它们的周长为 6 并且在几乎满足下界的意义上几乎是最优的。原则上，随机生成的高斯测量矩阵是“阶次最优的”。” 因此，我们比较了使用二进制阵列代码和高斯矩阵的基追踪公式的相变行为，并表明 (i) 两种情况下的相变边界之间基本上没有差异，以及 (ii) 基追踪的 CPU 时间使用二进制矩阵比使用高斯矩阵快数百倍，并且存储要求更少。因此，建议二进制矩阵是使用基追踪进行压缩感知的高斯矩阵的可行替代方案。