A stability result using the matrix norm to bound the permanent

TL;DR

This paper establishes a stability version of a bound on the permanent of a matrix in terms of its operator norm, showing that matrices close to extremal cases have nearly maximal permanent.

Contribution

It proves a stability result for the permanent bound, quantifying how the permanent decreases when the matrix deviates from extremal matrices.

Findings

Matrices close to extremal form have nearly maximal permanent.

The permanent is exponentially smaller unless most rows have entries close to the operator norm.

Provides quantitative bounds on how the permanent diminishes with deviation from extremal matrices.

Abstract

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 $\alpha, \beta > 0$, we show that $|\text{perm}(A)|$ is exponentially smaller than $\Vert A \Vert_2 ^n$ unless all but at most $\alpha n$ rows contain entries of modulus at least $\Vert A \Vert_2 (1 - \beta)$.