Szemerédi’s Regularity Lemma and Its Applications to Pairwise Clustering and Segmentation

Abstract

Szemerédi’s regularity lemma is a deep result from extremal graph theory which states that every graph can be well-approximated by the union of a constant number of random-like bipartite graphs, called regular pairs. Although the original proof was non-constructive, efficient (i.e., polynomial-time) algorithms have been developed to determine regular partitions for arbitrary graphs. This paper reports a first attempt at applying Szemerédi’s result to computer vision and pattern recognition problems. Motivated by a powerful auxiliary result which, given a partitioned graph, allows one to construct a small reduced graph which inherits many properties of the original one, we develop a two-step pairwise clustering strategy in an attempt to reduce computational costs while preserving satisfactory classification accuracy. Specifically, Szemerédi’s partitioning process is used as a preclustering step to substantially reduce the size of the input graph in a way which takes advantage of the strong notion of edge-density regularity. Clustering is then performed on the reduced graph using standard algorithms and the solutions obtained are then mapped back into the original graph to create the final groups. Experimental results conducted on standard benchmark datasets from the UCI machine learning repository as well as on image segmentation tasks confirm the effectiveness of the proposed approach.