Comparing Clusterings in Space

Abstract

This paper proposes a new method for comparing clusterings both partitionally and geometrically. Our approach is motivated by the following observation: the vast majority of previous techniques for comparing clusterings are entirely partitional, i.e., they examine assignments of points in set theoretic terms after they have been partitioned. In doing so, these methods ignore the spatial layout of the data, disregarding the fact that this information is responsible for generating the clusterings to begin with. We demonstrate that this leads to a variety of failure modes. Previous comparison techniques often fail to differentiate between significant changes made in data being clustered. We formulate a new measure for comparing clusterings that combines spatial and partitional information into a single measure using optimization theory. Doing so eliminates pathological conditions in previous approaches. It also simultaneously removes common limitations, such as that each clustering must have the same number of clusters or they are over identical datasets. This approach is stable, easily implemented, and has strong intuitive appeal. spatial properties as well as their cluster membership assignments. We view a clustering as a partition of a set of points located in a space with an associated distance function. This view is natural, since popular clustering algorithms, e.g., k-means, spectral clustering, affinity propagation, etc., take as input not only a collection of points to be clustered but also a distance function on the space in which the points lie. This distance function may be specified implicitly and it may be transformed by a kernel, but it must be defined one way or another and its properties are crucial to a clustering algorithm’s output. In contrast, almost all existing clustering comparison techniques ignore the distances between points, treating clusterings as partitions of disembodied atoms. While this approach has merit under some circumstances, it seems surprising to ignore the distance func