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 most comprehensive theoretical study of clustering stability by Ben-David and von Luxburg @cite_43 defined a notion of clustering stability and related it to properties of the decision boundaries of the algorithm. This is the starting point of the theoretical part of this work. We extend these notions to account for the considerably more complex Mapper construction.
@cite_43
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In this paper, we investigate stability-based methods for cluster model selection, in particular to select the number K of clusters. The scenario under consideration is that clustering is performed by minimizing a certain clustering quality function, and that a unique global minimizer exists. On the one hand we show that stability can be upper bounded by certain properties of the optimal clustering, namely by the mass in a small tube around the cluster boundaries. On the other hand, we provide counterexamples which show that a reverse statement is not true in general. Finally, we give some examples and arguments why, from a theoretic point of view, using clustering stability in a high sample setting can be problematic. It can be seen that distribution-free guarantees bounding the difference between the finite sample stability and the “true stability” cannot exist, unless one makes strong assumptions on the underlying distribution.