使用matrix norm来约束持久性的稳定性结果
A stability result using the matrix norm to bound the permanent
摘要：
我们证明了一个稳定的版本的一般结果，界定永久的矩阵在其运营商规范。更具体地，假设$A$是$\mathbb{C}$上的$n \times n$矩阵（分别为令$\mathcal{P}$表示在$\mathbb{C}$上的$n \times n$矩阵的集合（分别$\mathbb{R}$），其可以被写为置换矩阵乘以酉对角矩阵。则已知$A$的积式满足$|\text{perm}(A)| \leq \Vert A \Vert_{2} ^n$且等式当且仅当$A/ \Vert A \Vert_{2} \in \mathcal{P}$（其中$\Vert A \Vert_2$是$A$的算子$2$ -范数）.我们证明了这一结果的稳定性版本断言，除非$A$是非常接近（在特定意义上）这些极值矩阵之一，其永久性是指数小于（作为一个函数的$n$）比$\Vert A \Vert_2 ^n$。特别是，对于任何固定的$α, β> 0$，我们表明，$|\text{perm}(A)|$是指数小于$\Vert A \Vert_2 ^n$，除非所有，但最多$αn$行包含条目的模数至少$\Vert A \Vert_2 (1 - β)$。
英文摘要：
We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose $A$ is an $n \times n$ matrix over $\mathbb{C}$ (resp. $\mathbb{R}$), and let $\mathcal{P}$ denote the set of $n \times n$ matrices over $\mathbb{C}$ (resp. $\mathbb{R}$) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of $A$ satisfies $|\text{perm}(A)| \leq \Vert A \Vert_{2} ^n$ with equality iff $A/ \Vert A \Vert_{2} \in \mathcal{P}$ (where $\Vert A \Vert_2$ is the operator $2$-norm of $A$). We show a stability version of this result asserting that unless $A$ is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of $n$) than $\Vert A \Vert_2 ^n$. In particular, for any fixed $α, β> 0$, we show that $|\text{perm}(A)|$ is exponentially smaller than $\Vert A \Vert_2 ^n$ unless all but at most $αn$ rows contain entries of modulus at least $\Vert A \Vert_2 (1 - β)$.
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