Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper, we propose a new spectral method, graph matching by pairwise eigen-alignments (GRAMPA). Departing from prior spectral approaches that only compare top eigenvectors, or eigenvectors of the same order, GRAMPA first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, with weights given by a Cauchy kernel applied to the separation of the corresponding eigenvalues, then outputs a matching by a simple rounding procedure. The similarity matrix can also be interpreted as the solution to a regularized quadratic programming relaxation of the quadratic assignment problem. For the Gaussian Wigner model in which two complete graphs on n vertices have Gaussian edge weights with correlation coefficient \(1-\sigma ^2\), we show that GRAMPA exactly recovers the correct vertex correspondence with high probability when \(\sigma = O(\frac{1}{\log n})\). This matches the state of the art of polynomial-time algorithms and significantly improves over existing spectral methods which require \(\sigma \) to be polynomially small in n. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency. Universality results, including similar guarantees for dense and sparse Erdős–Rényi graphs, are deferred to a companion paper.

图匹配旨在找到两个未标记图之间的顶点对应关系，以最大化总边权重相关性。这相当于解决计算上难以处理的二次分配问题。在本文中，我们提出了一种新的谱方法，即通过成对特征对齐 (GRAMPA) 进行图匹配。与仅比较顶部特征向量或相同阶的特征向量的先前光谱方法不同，GRAMPA 首先将相似性矩阵构造为所有之间的外积的加权和两个图的特征向量对，由柯西核给出的权重应用于相应特征值的分离，然后通过简单的舍入过程输出匹配。相似度矩阵也可以解释为二次分配问题的正则化二次规划松弛的解决方案。对于高斯维格纳模型，其中n个顶点上的两个完整图具有相关系数为\(1-\sigma ^2\)的高斯边权重，我们表明 GRAMPA 在\(\sigma = O(\frac{1}{\log n})\)。这与多项式时间算法的最新技术相匹配，并显着改进了现有的频谱方法，这些方法需要\(\sigma \)在n中是多项式小。在统计准确性和计算效率方面，GRAMPA 的优势也体现在各种合成和真实数据集上。普遍性结果，包括对密集和稀疏 Erdős-Rényi 图的类似保证，推迟到配套论文。