Mapper is an unsupervised machine learning algorithm generalising the notion of clustering to obtain a geometric description of a dataset. The procedure splits the data into possibly overlapping bins which are then clustered. The output of the algorithm is a graph where nodes represent clusters and edges represent the sharing of data points between two clusters. However, several parameters must be selected before applying Mapper and the resulting graph may vary dramatically with the choice of parameters. We define an intrinsic notion of Mapper instability that measures the variability of the output as a function of the choice of parameters required to construct a Mapper output. Our results and discussion are general and apply to all Mapper-type algorithms. We derive theoretical results that provide estimates for the instability and suggest practical ways to control it. We provide also experiments to illustrate our results and in particular we demonstrate that a reliable candidate Mapper output can be identified as a local minimum of instability regarded as a function of Mapper input parameters.
The work of Ackerman and Ben-David @cite_50 studied clustering quality measures rather than the clustering functions, which provides a richer setting in which an alternative to Kleinberg's axioms can be consistently stated.
@cite_50
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Aiming towards the development of a general clustering theory, we discuss abstract axiomatization for clustering. In this respect, we follow up on the work of Kleinberg, ([1]) that showed an impossibility result for such axiomatization. We argue that an impossibility result is not an inherent feature of clustering, but rather, to a large extent, it is an artifact of the specific formalism used in [1]. As opposed to previous work focusing on clustering functions, we propose to address clustering quality measures as the object to be axiomatized. We show that principles like those formulated in Kleinberg's axioms can be readily expressed in the latter framework without leading to inconsistency. A clustering-quality measure (CQM) is a function that, given a data set and its partition into clusters, returns a non-negative real number representing how strong or conclusive the clustering is. We analyze what clustering-quality measures should look like and introduce a set of requirements (axioms) for such measures. Our axioms capture the principles expressed by Kleinberg's axioms while retaining consistency. We propose several natural clustering quality measures, all satisfying the proposed axioms. In addition, we analyze the computational complexity of evaluating the quality of a given clustering and show that, for the proposed CQMs, it can be computed in polynomial time.