Asymmetric Cheeger cut and application to multi-class unsupervised clustering

Cheeger cut has recently been shown to provide excellent classification results for two classes. Whereas the classical Cheeger cut favors a 50-50 partition of the graph, we present here an asymmetric variant of the Cheeger cut which favors, for example, a 10-90 partition. This asymmetric Cheeger cut provides a powerful tool for unsupervised multi-class partitioning. We use it in recursive bipartitioning to detach one after the other each of the classes. This asymmetric recursive algorithm handles equally well any number of classes, as opposed to symmetric recursive bipartitioning which is naturally better suited for 2 classes. We obtain an error classification rate of 2.35% and 4.07% for MNIST and USPS benchmark datasets respectively, drastically improving the former 11.7% and 13% error rate obtained in the literature with symmetric Cheeger cut bipartitioning algorithms.