Neural Graph Matching Network: Learning Lawler's Quadratic Assignment Problem with Extension to Hypergraph and Multiple-graph Matching

Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawler's Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a vertex classification task. The association graph is learned by an embedding network for vertex classification, followed by Sinkhorn normalization and a cross-entropy loss for end-to-end learning. We further improve the embedding model on association graph by introducing Sinkhorn based constraint, and dummy nodes to deal with outliers. To our best knowledge, this is the first network to directly learn with the general Lawler's QAP. In contrast, state-of-the-art deep matching methods focus on the learning of node and edge features in two graphs respectively. We also show how to extend our network to hypergraph matching, and matching of multiple graphs. Experimental results on both synthetic graphs and real-world images show our method outperforms. For pure QAP tasks on synthetic data and QAPLIB, our method can surpass spectral matching and RRWM, especially on challenging problems.