Although graphs are widely used in a myriad of machine learning tasks, they are limited to representing pairwise interactions. By contrast, in many real-world applications the entities engage in higher-order relations. Such relations can be modeled by hypergraphs, where the notion of an edge is generalized to a hyperedge that can connect more than two vertices. Traditional hypergraph models treat all the vertices in a hyperedge equally while in practice these vertices might contribute differently to the hyperedge. To deal with such cases, edge-dependent vertex weights (EDVWs) are introduced into hypergraphs which are able to reflect different importance of vertices within the same hyperedge.

In this thesis, I study several fundamental problems considering the hypergraph model with EDVWs. First, I develop valid Laplacian matrices for this hypergraph model through random walks defined on vertices and hyperedges and incorporating EDVWs, based on which I propose spectral partitioning algorithms for co-clustering vertices and hyperedges. Second, I develop a framework for incorporating EDVWs into hypergraph cut problems via introducing a new class of hyperedge splitting functions which are both submodular and dependent on EDVWs. I also generalize existing reduction as well as sparsification techniques to our setting. Finally, I define p-Laplacians for this hypergraph model and focus on the p=1 case. I propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the 1-Laplacian and then use this eigenvector to cluster vertices in order to achieve better performance than traditional spectral clustering based on the 2-Laplacian.