of Half-Lives EigenFlows for Spectral Clustering
Chakra Chennubhotla & Allan D. Jepson Department of Computer Science, University of Toronto, Canada M5S 3H5 chakra,jepson @cs.toronto.edu
Abstract
Using a Markov chain perspective of spectral clustering we present an algorithm to automatically nd the number of stable clusters in a dataset. The Markov chains behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenows along with their halives. An eigenow describes the ow of probability mass due to the Markov chain, and it is characterized by its eigenvalue, or equivalently, by the halife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenow and innite half-life. The key insight in this paper is that bottlenecks between weakly coupled clusters can be identied by computing the sensitivity of the eigenows halife to variations in the edge weights. We propose a novel E IGEN C UTS algorithm to perform clustering that removes these identied bottlenecks in an iterative fashion.
1 Introduction
We consider partitioning a weighted undirected graph corresponding to a given dataset into a set of discrete clusters. Ideally, the vertices (i.e. datapoints) in each cluster should be connected with high-afnity edges, while different clusters are either not connected or are connnected only by a few edges with low afnity. The practical problem is to identify these tightly coupled clusters, and cut the inter-cluster edges. Many techniques have been proposed for this problem, with some recent success being obtained through the use of spectral methods (see, for example, [2, 4, 5, 11, 12]). Here we use the random walk formulation of [4], where the edge weights are used to construct a Markov transition probability matrix, . This matrix denes a random walk on the graph to be partitioned. The eigenvalues and eigenvectors of provide the basis for deciding on a particular segmentation. In particular, it has been shown that for weakly coupled clusters, the leading eigenvectors of will be roughly piecewise constant [4, 13, 5]. This result motivates many of the current spectral clustering algorithms. For example in [5], the number of clusters must be known a priori, and the -means algorithm is used on the leading eigenvectors of in an attempt to identify the appropriate piecewise constant regions. In this paper we investigate the form of the leading eigenvectors of the Markov matrix . Using some simple image segmentation examples we conrm that the leading eigenvectors of ar...
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