Relational Learning with Hypergraphs

Relational learning has received extensive attentions in recent years since a huge amount of data is generated every day in the cyber-space andmost of them is organized by the relations between entities. Themain tasks of the relational learning include discovering the communities of entities, classifying the entities, and make predictions of possible new relations. Since the graph is a natural representation of pairwise relations, these tasks have been widely studied using the graphs. In this work, we examine the relational learning tasks in the framework of hypergraph which is an extension of the graph. In a graph, an edge could connect exactly two vertices, while in a hypergraph a hyperedge could connect any number of vertices. This extension from graph to hypergraph allows us to represent the higher-order relations such as the co-occurrence relation. The existing works of hypergraph learning mainly focus on the so-called “vertex expansion” where the hypergraph is transformed into a graph that shares the same set of vertices with the hypergraph. With different weighting functions used in the transformation, the resulting graph would have different structures. The spectral graph theory provides us a powerful tool to analyze the graph structures. It has been shown that one can use the eigenvectors of the graph Laplacian to discover clusters of vertices in the graph. Therefore, the spectral techniques are also adapted to the graph transformed from the hypergraph, which serves as the main ingredient of the clustering and classification algorithms. We show that a special vertex expansion called the normalized hypergraph cut (NHC) can be also used in the link prediction task to rank the possible relations that would appear in the future. In fact, the NHC expansion is able to produce a latent factor space where each entity is represented by a vector and all the vectors form approximately orthogonal clusters. On the other hand, instead of taking the vertex-centric view in the vertex expansions, we turn to the hyperedge-centric view and develop the “hyperedge expansion” that reflects another category of objective functions defined on the hyperedges. We show that the hyperedge expansion objectives can be attained by computing the eigenvectors of the Laplacian of a directed auxiliary graph, and this eigen-decomposition is equivalent to a quadratic eigenvalue problem (QEP). Based on the analysis of the above eigen-decomposition, we present the clustering and classification algorithms with the hyperedge expansion. All the approaches developed in this work are compared with state-of-the-art methods in