Tangles: From Weak to Strong Clustering

We introduce a new approach to clustering by using tangles, a tool that originates in mathematical graph theory. Given a collection of "weak partitions" of a data set, tangles provide a framework to aggregate these weak partitions such that they "point in the direction of a cluster". As a result, a cluster is softly characterized by a set of consistent pointers. This mechanism provides a highly flexible way of solving soft clustering problems in a variety of setups, ranging from questionnaires over community detection in graphs to clustering points in metric spaces. Conceptually, tangles have many intriguing properties: (1) Similar to boosting, which combines many weak classifiers to a strong classifier, tangles provide a formal way to combine many weak partitions to obtain few strong clusters. (2) In terms of computational complexity, tangles allow us to use simple, fast algorithms to produce the weak partitions. The complexity of identifying the strong partitions is dominated by the number of weak partitions, not the number of data points, leading to an interesting trade-off between the two. (3) If the weak partitions are interpretable, so are the strong partitions induced by the tangles, resulting in one of the rare algorithms to produce interpretable clusters. (4) The output of tangles is of a hierarchical nature, inducing the notion of a soft dendrogram that can be helpful in data visualization.