arXiv 2103.093832021-03-17math.STcs.ITmath.COmath.ITmath.PRstat.MLstat.TH

种植匹配问题：锐阈值和无限阶相变

The planted matching problem: Sharp threshold and infinite-order phase transition

Jian Ding, Yihong Wu, Jiaming Xu, Dana Yang

摘要：

我们研究的问题，重建一个完美匹配的$M^*$隐藏在随机加权$n\times n$二部图。边集合以概率$d/n$独立地包括$M^*$中的每个节点对和不在$M^*$中的每个$n(n-1)$节点对。每个边$e$的权重独立地从分布$\mathcal{P}$（如果是$e \in M^*$）和$\mathcal{Q}$（如果是$e \notin M^*$）中提取。我们表明，如果$\sqrt{d} B(\mathcal{P},\mathcal{Q}) \le 1$，其中$B(\mathcal{P},\mathcal{Q})$代表巴塔查里亚系数，重建误差（平均分数的错误分类边缘）的最大似然估计$M^*$收敛到$0$作为$n\to \infty$。相反，如果$\sqrt{d} B(\mathcal{P},\mathcal{Q}) \ge 1+ε$对于任意小的常数$ε>0$，则在稀疏和密集模型下，任何估计量的重建误差都与$0$有界，解决了[Moharrami et al. 2019，Semerjian et al. 2020]中的猜想。此外，在具有$d=n$，$\mathcal{P}=\exp(λ)$和$\mathcal{Q}=\exp(1/n)$的完全指数加权图的特殊情况下，尖锐阈值简化为$λ=4$，我们证明了当$λ\le 4-ε$时，最佳重建误差为$\exp\left( - Θ(1/\sqrtε) \right)$，从而证实了[Semerjian et al. 2020]中所描述的无限阶相变。

英文摘要：

We study the problem of reconstructing a perfect matching $M^*$ hidden in a randomly weighted $n\times n$ bipartite graph. The edge set includes every node pair in $M^*$ and each of the $n(n-1)$ node pairs not in $M^*$ independently with probability $d/n$. The weight of each edge $e$ is independently drawn from the distribution $\mathcal{P}$ if $e \in M^*$ and from $\mathcal{Q}$ if $e \notin M^*$. We show that if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \le 1$, where $B(\mathcal{P},\mathcal{Q})$ stands for the Bhattacharyya coefficient, the reconstruction error (average fraction of misclassified edges) of the maximum likelihood estimator of $M^*$ converges to $0$ as $n\to \infty$. Conversely, if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \ge 1+ε$ for an arbitrarily small constant $ε>0$, the reconstruction error for any estimator is shown to be bounded away from $0$ under both the sparse and dense model, resolving the conjecture in [Moharrami et al. 2019, Semerjian et al. 2020]. Furthermore, in the special case of complete exponentially weighted graph with $d=n$, $\mathcal{P}=\exp(λ)$, and $\mathcal{Q}=\exp(1/n)$, for which the sharp threshold simplifies to $λ=4$, we prove that when $λ\le 4-ε$, the optimal reconstruction error is $\exp\left( - Θ(1/\sqrtε) \right)$, confirming the conjectured infinite-order phase transition in [Semerjian et al. 2020].

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。

摘要：

英文摘要：

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