Higher-order clustering with non-uniform hypergraphs

Abstract

In general terms, clustering can be defined as the problem of organizing a finite set of elements into groups in such a way that similar elements belong to the same cluster while dissimilar elements are assigned to different clusters. Most of the approaches available in literature focuses on pairwise clustering, assuming that the similarities between objects are strictly pairwise and can be represented using a simple weighted graph; however, in many situations this assumption is limiting: higher-order relations seem to be more appropriate to reduce potential information loss. When clustering takes into account these higher-order similarities it assumes the name "hypergraph clustering", since the data that needs to be grouped can be represented using a hypergraph: the nodes are the objects to be clustered while the hyperedges represent high-order similarities. Another assumption is however in place: hyperedges must have the same fixed cardinality. The goal of the present thesis is to identify and test a hypergraph clustering approach that takes into account the different higher-order relations existing between sets of objects, dismissing the two previous assumptions: starting from the observations of Pelillo et al. on clustering using uniform hypergraphs, this work explores the possible implications of clustering using non-uniform hypergraphs, providing a detailed examination of several test cases.