Sparse binary matrices are of great interest in the field of sparse recovery, nonnegative compressed sensing, statistics in networks, and theoretical computer science. This class of matrices makes it possible to perform signal recovery with lower storage costs and faster decoding algorithms. In particular, Bernoulli$(p)$ matrices formed by independent identically distributed (i.i.d.) Bernoulli$(p)$ random variables are of practical relevance in the context of noise-blind recovery in nonnegative compressed sensing. In this work, we investigate the robust nullspace property of Bernoulli$(p)$ matrices. Previous results in the literature establish that such matrices can accurately recover $n$-dimensional $s$-sparse vectors with $m=O\left(\frac{s}{c(p)}\log\frac{en}{s}\right)$ measurements, where $c(p) \le p$ is a constant dependent only on the parameter $p$. These results suggest that in the sparse regime, as $p$ approaches zero, the (sparse) Bernoulli$(p)$ matrix requires significantly more measurements than the minimal necessary, as achieved by standard isotropic subgaussian designs. However, we show that this is not the case. Our main result characterizes, for a wide range of sparsity levels $s$, the smallest $p$ for which sparse recovery can be achieved with the minimal number of measurements. We also provide matching lower bounds to establish the optimality of our results and explore connections with the theory of invertibility of discrete random matrices and integer compressed sensing.

翻译：稀疏二进制矩阵在稀疏恢复、非负压缩感知、网络统计及理论计算机科学领域具有重要研究价值。该类矩阵能够以更低的存储成本和更快的解码算法实现信号恢复。特别地，由独立同分布伯努利$(p)$随机变量构成的伯努利$(p)$矩阵，在非负压缩感知的噪声盲恢复场景中具有实际意义。本研究探讨了伯努利$(p)$矩阵的鲁棒零空间性质。现有文献结果表明，此类矩阵可通过$m=O\left(\frac{s}{c(p)}\log\frac{en}{s}\right)$次测量精确恢复$n$维$s$-稀疏向量，其中$c(p) \le p$是仅依赖于参数$p$的常数。这些结果表明在稀疏区域中，当$p$趋近于零时，（稀疏）伯努利$(p)$矩阵所需的测量次数显著超过标准各向同性亚高斯设计所能达到的最小必要测量次数。然而，我们证明实际情况并非如此。我们的主要成果在于：针对广泛的稀疏度范围$s$，刻画了能够以最小测量次数实现稀疏恢复的最小$p$值。同时，我们给出了匹配的下界以证明结果的优化性，并探讨了与离散随机矩阵可逆性理论及整数压缩感知理论的关联。