Clustering is difficult only when it does not matter by Amit Daniely, Nati Linial, Michael Saks.
Abstract:
Numerous papers ask how difficult it is to cluster data. We suggest that the more relevant and interesting question is how difficult it is to cluster data sets {\em that can be clustered well}. More generally, despite the ubiquity and the great importance of clustering, we still do not have a satisfactory mathematical theory of clustering. In order to properly understand clustering, it is clearly necessary to develop a solid theoretical basis for the area. For example, from the perspective of computational complexity theory the clustering problem seems very hard. Numerous papers introduce various criteria and numerical measures to quantify the quality of a given clustering. The resulting conclusions are pessimistic, since it is computationally difficult to find an optimal clustering of a given data set, if we go by any of these popular criteria. In contrast, the practitioners’ perspective is much more optimistic. Our explanation for this disparity of opinions is that complexity theory concentrates on the worst case, whereas in reality we only care for data sets that can be clustered well.
We introduce a theoretical framework of clustering in metric spaces that revolves around a notion of “good clustering”. We show that if a good clustering exists, then in many cases it can be efficiently found. Our conclusion is that contrary to popular belief, clustering should not be considered a hard task.
Considering that clustering is a first step towards merging, you will find the following encouraging:
From the practitioner’s viewpoint, “clustering is either easy or pointless” &emdash; that is, whenever the input admits a good clustering, finding it is feasible. Our analysis provides some support to this view.
I would caution that the authors are working with metric spaces.
It isn’t clear to me that clustering based on values in non-metric spaces would share the same formal characteristics.
Comments or pointers to work on clustering in non-metric spaces?