谱图匹配与正则二次松弛II：Erdens-Rényi图与普适性
Spectral Graph Matching and Regularized Quadratic Relaxations II: Erdős-Rényi Graphs and Universality
摘要：
我们分析了一种新的谱图匹配算法，GRAph匹配成对特征对齐（GRAMPA），恢复两个未标记的，边缘相关的加权图之间的潜在顶点对应关系。扩展的确切恢复保证建立在同伴纸高斯权重，在这项工作中，我们证明了这些保证的普遍性一般相关的维格纳模型。特别地，对于两个边相关系数为$1-σ^2$且平均度至少为$\operatorname{polylog}(n)$的Erdens-Rényi图，我们证明了当$σ\lesssim 1/\operatorname{polylog}(n)$ .此外，我们建立了一个类似的保证GRAMPA的一个变种，对应于一个更紧密的二次规划放松的二次指派问题。我们的分析利用了一个预解式表示的GRAMPA相似矩阵和本地法律的稀疏维格纳矩阵的预解式。
英文摘要：
We analyze a new spectral graph matching algorithm, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. Extending the exact recovery guarantees established in the companion paper for Gaussian weights, in this work, we prove the universality of these guarantees for a general correlated Wigner model. In particular, for two Erdős-Rényi graphs with edge correlation coefficient $1-σ^2$ and average degree at least $\operatorname{polylog}(n)$, we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when $σ\lesssim 1/\operatorname{polylog}(n)$. Moreover, we establish a similar guarantee for a variant of GRAMPA, corresponding to a tighter quadratic programming relaxation of the quadratic assignment problem. Our analysis exploits a resolvent representation of the GRAMPA similarity matrix and local laws for the resolvents of sparse Wigner matrices.
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