Instability of the Betti Sequence for Persistent Homology and a\n Stabilized Version of the Betti Sequence

Topological Data Analysis (TDA), a relatively new field of data analysis, has\nproved very useful in a variety of applications. The main persistence tool from\nTDA is persistent homology in which data structure is examined at many scales.\nRepresentations of persistent homology include persistence barcodes and\npersistence diagrams, both of which are not straightforward to reconcile with\ntraditional machine learning algorithms as they are sets of intervals or\nmultisets. The problem of faithfully representing barcodes and persistent\ndiagrams has been pursued along two main avenues: kernel methods and\nvectorizations. One vectorization is the Betti sequence, or Betti curve,\nderived from the persistence barcode. While the Betti sequence has been used in\nclassification problems in various applications, to our knowledge, the\nstability of the sequence has never before been discussed. In this paper we\nshow that the Betti sequence is unstable under the 1-Wasserstein metric with\nregards to small perturbations in the barcode from which it is calculated. In\naddition, we propose a novel stabilized version of the Betti sequence based on\nthe Gaussian smoothing seen in the Stable Persistence Bag of Words for\npersistent homology. We then introduce the normalized cumulative Betti sequence\nand provide numerical examples that support the main statement of the paper.\n