Persistent homology has emerged as one of the most important tools in topological data analysis. Recent results on the stability of the persistence diagram, describing the appearance and disappearance of topological features, allow us to investigate the behavior of a function on an underlying space with only mild assumptions and a finite point-sample. First, we apply these results to the problem of clustering data sampled from some underlying distribution. Under the appropriate conditions, we can provably recover the correct number of clusters as well as show some spatial stability. Combining these results, we obtain a topological soft-clustering algorithm. The algorithm can be multiple times with pertubations to the data set to obtain a series of clusterings. The theoretical results guarantee that we can find the cluster correspondances between runs. We present several possible algorithms arising from this general scheme and show preliminary results under various types of pertubations.
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