Probability Seminar
Thursday, September 19, 2019, 3:15pm, 119 Physics
Jiaming Xu (Fuqua)
Spectral graph matching and regularized quadratic relaxations

Abstract:

Given two unlabeled, edge-correlated graphs on the same set of vertices, we study the "graph matching" problem of identifying the unknown mapping from vertices of the first graph to those of the second. This amounts to solving a computationally intractable quadratic assignment problem. We propose a new spectral method, which computes eigendecomposition of the two graph adjacency matrices and returns a matching based on the pairwise alignments between all eigenvectors of the first graph with all eigenvectors of the second. Each alignment is inversely weighted by the distance between the corresponding eigenvalues. This spectral method can be equivalently viewed as solving a regularized quadratic programming relaxation of the quadratic assignment problem. We show that for a correlated Erdos-Renyi model, this method can return the exact matching with high probability if the two graphs differ by at most a 1/polylog(n) fraction of edges, both for dense graphs and for sparse graphs with at least polylog(n) average degree. Our analysis exploits local laws for the resolvents of sparse Wigner matrices. Based on joint work with Zhou Fan, Cheng Mao, Yihong Wu, all at Yale. Preprints available at https://arxiv.org/abs/1907.08880 and https://arxiv.org/abs/1907.08883.

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