Optimization algorithms for graph learning and graph matching problems

Abstract

Graph data provides information about data points individually, as well as information about the relationships between these data points. Based on the assumption that nodes are related to their neighbors, methods to solve problems on graph data leverage the graph structure in addition to the node feature information to make predictions. Many applications can be represented as graphs, including social networks, citation networks, and biological structures. Our work proposes methods for two tasks on graph data: the node classification task and the graph matching task. The node classification task assigns labels to nodes in a graph based on the node's own feature information, as well as the feature information of its neighboring nodes. The graph matching task finds pairwise correspondences between two sets of nodes. Each set of nodes comes from the same graph, however the matrix representations of the graph structure are different due to the arbitrary order of the node labeling. We first motivate our work by discussing specific applications of both of these types of problems. We then propose several methods for solving the node classification task by graph convolutional networks (GCNs). They combine different momentum-based optimizers with an existing GCN training method. We provide convergence guarantees for these methods, and provide numerical results on several benchmark datasets that show the benefit of using these momentum-based optimizers over the classic stochastic gradient descent. Next, we propose algorithms for solving the graph matching problem that relax the binary constraint and introduce a penalty term. We derive the primal and dual variable updates for the Alternating Direction Method of Multipliers to solve this problem. Finally, we show that our methods achieve comparable accuracy and superior efficiency in terms of running time over two other algorithms on randomly generated graphs and subgraphs of two benchmark citation network graphs.