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Topological Data Analysis (TDA), a relatively new field of data analysis, hasproved very useful in a variety of applications. The main persistence tool fromTDA is persistent homology in which data structure is examined at many scales.Representations of persistent homology include persistence barcodes andpersistence diagrams, both of which are not straightforward to reconcile withtraditional machine learning algorithms as they are sets of intervals ormultisets. The problem of faithfully representing barcodes and persistentdiagrams has been pursued along two main avenues: kernel methods andvectorizations. One vectorization is the Betti sequence, or Betti curve,derived from the persistence barcode. While the Betti sequence has been used inclassification problems in various applications, to our knowledge, thestability of the sequence has never before been discussed. In this paper weshow that the Betti sequence is unstable under the 1-Wasserstein metric withregards to small perturbations in the barcode from which it is calculated. Inaddition, we propose a novel stabilized version of the Betti sequence based onthe Gaussian smoothing seen in the Stable Persistence Bag of Words forpersistent homology. We then introduce the normalized cumulative Betti sequenceand provide numerical examples that support the main statement of the paper.

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