Better Guarantees for Sparsest Cut Clustering

The field of approximation algorithms for clustering is a very active one and a large number of algorithms have been developed for clustering objectives such as k-median, min-sum, and sparsest cut clustering. For most of these objectives, the approximation guarantees do not match the known hardness results, and much effort is spent on obtaining tighter approximation guarantees [1, 4, 5, 8, 6, 9, 10]. However, for many practical clustering problems such as clustering proteins by function, or clustering images by subject, there is some unknown correct “target” clustering; in such cases the pairwise information is merely based on some heuristics and the real goal is to achieve low error on the data. In these settings, the implicit hope is that approximately optimizing objective functions such as those mentioned above will in fact produce a clustering of low error, i.e., a clustering that is close pointwise to the truth. Formally, for a set of n data points the error of a clustering the error of a clustering C′ = {C ′ 1, ..., C ′ k} with respect to target clustering C = {C1, ..., Ck} is the fraction of points on which C and C′ disagree under the optimal matching of clusters in C to clusters in C′, i.e.