An information-theoretic derivation of min-cut based clustering

Min-cut clustering, based on minimizing one of two heuristic cost-functions proposed by Shi and Malik, has spawned tremendous research, both analytic and algorithmic, in the graph partitioning and image segmentation communities over the last decade. It is however unclear if these heuristics can be derived from a more general principle facilitating generalization to new problem settings. Motivated by an existing graph partitioning framework, we derive relationships between optimizing relevance information, as defined in the Information Bottleneck method, and the regularized cut in a K-partitioned graph. For fast mixing graphs, we show that the cost functions introduced by Shi and Malik can be well approximated as the rate of loss of predictive information about the location of random walkers on the graph. For graphs generated from a stochastic algorithm designed to model community structure, the optimal information theoretic partition and the optimal min-cut partition are shown to be the same with high probability.