离散随机矩阵的稀疏恢复性质
Sparse recovery properties of discrete random matrices
摘要：
受压缩感知问题的启发，我们确定了随机$n\times d$ $\pm 1$矩阵$M_{n,d}$关于“每个$s$列是线性无关“的阈值行为。特别是，我们表明，对于每一个$0<δ<1$和$s=(1-δ)n$，如果$d\leq n^{1+1/2(1-δ)-o(1)}$，那么有很高的概率$M_{n,d}$的每一个$s$列是线性无关的，如果$d\geq n^{1+1/2(1-δ)+o(1)}$，那么有很高的概率有一些$s$线性相关的列。
英文摘要：
Motivated by problems from compressed sensing, we determine the threshold behavior of a random $n\times d$ $\pm 1$ matrix $M_{n,d}$ with respect to the property "every $s$ columns are linearly independent". In particular, we show that for every $0<δ<1$ and $s=(1-δ)n$, if $d\leq n^{1+1/2(1-δ)-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-δ)+o(1)}$ then with high probability there are some $s$ linearly dependent columns.
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