Clustering with $r$-regular graphs

Abstract

In this paper, we present a novel graph-based clustering method, where we decompose a (neighborhood) graph into (disjoint) $r$-regular graphs followed by further refinement through optimizing the normalized cluster utility. We solve the $r$-regular graph decomposition using a linear programming. However, this simple decomposition suffers from inconsistent edges if clusters are not well separated. We optimize the normalized cluster utility in order to eliminate inconsistent edges or to merge similar clusters into a group within the principle of minimal $K$-cut. The method is especially useful in the presence of noise and outliers. Moreover, the method detects the number of clusters within a pre-specified range. Numerical experiments with synthetic and UCI data sets, confirm the useful behavior of the method.

Introduction

Clustering is a fundamental technique in exploratory data analysis, which is often encountered into many applications such as data mining, image segmentation, information retrieval, and bioinformatics [9]. There exist many successful clustering algorithms, however, clustering is still a difficult problem because the notion of clusters is strongly dependent on the context as well as the purpose of clustering. The main issues that are unsolved and controversial in clustering, can be summarized as follows [10]. First, it is data-dependent to select appropriate measure of similarity or dissimilarity between data points. There is no guidelines which allow us to choose the best one among diverse measures. Second, there exist widely acceptable conceptual definitions of a cluster, but it is difficult to derive an operational definition leading to a concrete algorithm from the conceptual one. The operational definition is also related to selecting the optimal number of clusters which is usually unknown and is difficult to estimate in real world applications. Third, the quality of clustering is very sensitive to background noise or outliers. Therefore, it is important to remove or to discriminate them from actual clusters, for robust clustering. Fourth, many clustering algorithms, which are based on minimizing the cost of error iteratively, suffer from getting trapped in local minima. Last, the hyperparameters such as the kernel width in spectral clustering are not easy to tune manually [20].

In this paper, we present a novel $r$-regular graph-based clustering algorithm which can alleviate the aforementioned problems. In Section 2, we give a review of graph-based clustering algorithms. In Section 3, we first define an operational definition of a cluster in terms of an underlying graph structure and the dissimilarity between vertices. The optimal clusters are determined by maximizing the normalized cluster utility. We emphasize that the graph structure of $r$-regular graphs plays an important role in eliminating inconsistent edges and in calculating the normalized cluster utility. In Section 4, we explain how to decompose a graph into disjoint $r$-regular graphs. Our proposed clustering algorithm is presented in Section 5, incorporating the decomposed $r$-regular graph into the definition of a cluster. Numerical experimental results with synthetic and UCI data sets are provided in Section 6, showing the useful behavior of our method. Finally conclusions are drawn and discussions are given in Section 7.