相关高斯Wigner模型下基于投影幂方法的种子图匹配
Seeded graph matching for the correlated Gaussian Wigner model via the projected power method
AI总结：
本文研究相关高斯Wigner模型下的种子图匹配问题，证明了投影幂方法在种子足够接近真实匹配时能以高概率在O(log n)次迭代内恢复匹配，将已有稀疏模型分析扩展到稠密模型。
摘要：
在图匹配问题中，我们观察到两个图$G,H$，目标是找到它们顶点之间的一个分配（或匹配），使得某种边一致性度量最大化。本文假设观测到的图对$G,H$来自相关高斯Wigner（CGW）模型——一种流行的相关加权图模型——其中$G$和$H$的邻接矩阵元素是独立的高斯随机变量，$G$的每条边与$H$的一条边（由未知匹配决定）相关，边相关性由参数$σ\in [0,1)$描述。在本文中，我们分析了投影幂方法（PPM）作为一种种子图匹配算法的性能，在该设置中我们获得一个初始的部分正确匹配（称为种子）作为辅助信息。我们证明，如果种子与真实匹配足够接近，则以高概率，PPM会迭代地改进种子并在$\mathcal{O}(\log n)$次迭代内恢复真实匹配（部分或完全）。我们的结果证明PPM即使在常数$σ$的情况下也能工作，从而将（Mao et al. 2023）针对稀疏相关Erdos-Renyi（CER）模型的分析扩展到（稠密）CGW模型。作为我们分析的一个副产品，我们看到PPM框架推广了一些用于种子图匹配的最先进算法。我们用合成数据上的数值实验支持和补充了我们的理论发现。
英文摘要：
In the \emph{graph matching} problem we observe two graphs $G,H$ and the goal is to find an assignment (or matching) between their vertices such that some measure of edge agreement is maximized. We assume in this work that the observed pair $G,H$ has been drawn from the Correlated Gaussian Wigner (CGW) model -- a popular model for correlated weighted graphs -- where the entries of the adjacency matrices of $G$ and $H$ are independent Gaussians and each edge of $G$ is correlated with one edge of $H$ (determined by the unknown matching) with the edge correlation described by a parameter $σ\in [0,1)$. In this paper, we analyse the performance of the \emph{projected power method} (PPM) as a \emph{seeded} graph matching algorithm where we are given an initial partially correct matching (called the seed) as side information. We prove that if the seed is close enough to the ground-truth matching, then with high probability, PPM iteratively improves the seed and recovers the ground-truth matching (either partially or exactly) in $\mathcal{O}(\log n)$ iterations. Our results prove that PPM works even in regimes of constant $σ$, thus extending the analysis in (Mao et al. 2023) for the sparse Correlated Erdos-Renyi(CER) model to the (dense) CGW model. As a byproduct of our analysis, we see that the PPM framework generalizes some of the state-of-art algorithms for seeded graph matching. We support and complement our theoretical findings with numerical experiments on synthetic data.
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