Isoperimetric cut on a directed graph

In this paper, we propose a novel probabilistic view of the spectral clustering algorithm. In our framework, the spectral clustering algorithm can be viewed as assigning class labels to samples to minimize the Bayes classification error rate by using a kernel density estimator (KDE). From this perspective, we propose to construct directed graphs using variable bandwidth KDEs. Such a variable bandwidth KDE based directed graph has the advantage that it encodes the local density information of the data in the graph edge weights. In order to cluster vertices of the directed graph, we develop a directed graph partitioning algorithm which optimizes a random walk isoperimetric ratio. The partitioning result can be obtained efficiently by solving a system of linear equations. We have applied our algorithm to several benchmark data sets and obtained promising results.