arXiv 1702.040502017-02math.PR

变形随机矩形矩阵的最小奇异值

The smallest singular value of deformed random rectangular matrices

Fan Yang

备注：: 23 pages

摘要：

证明了乘加变形随机矩形矩阵的最小奇异值的一个估计。假设对于某个常数$Λ\ge 1$，有$n\le N \le M \le ΛN$。设$X$是一个$M\times n$随机矩阵，其元素独立同分布，均值为零，方差为单位，矩为任意高。令$T$是$N\times M$确定性矩阵，对于某个常数$c>0$，具有可比奇异值$c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$。令$A$是具有$\|A\|=O(\sqrt{N})$的$N\times n$确定性矩阵。然后证明了对于任意的$ε>0$，$TX-A$的最小奇异值以很高的概率大于$N^{-ε}(\sqrt{N}-\sqrt{n-1})$.如果我们进一步假设$X$的项具有亚高斯衰减，则$TX-A$的最小奇异值至少具有高概率的$\sqrt{N}-\sqrt{n-1}$阶，这是本质上最优的估计。

英文摘要：

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le ΛN$ for some constant $Λ\ge 1$. Let $X$ be an $M\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$ deterministic matrix with comparable singular values $c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$ deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any $ε>0$, the smallest singular value of $TX-A$ is larger than $N^{-ε}(\sqrt{N}-\sqrt{n-1})$ with high probability. If we assume further the entries of $X$ have subgaussian decay, then the smallest singular value of $TX-A$ is at least of the order $\sqrt{N}-\sqrt{n-1}$ with high probability, which is an essentially optimal estimate.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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