Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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INSTABILITY OF THE BETTI SEQUENCE FOR PERSISTENT HOMOLOGY AND A STABILIZED VERSION OF THE BETTI SEQUENCE

  • JOHNSON, MEGAN (DEPARTMENT OF MATHEMATICS, UNIVERSITY AT BUFFALO, THE STATE UNIVERSITY OF NEW YORK) ;
  • JUNG, JAE-HUN (DEPARTMENT OF MATHEMATICS & POSTECH MATHEMATICAL INSTITUTE FOR DATA SCIENCE (MINDS), POHANG UNIVERSITY OF SCIENCE AND TECHNOLOGY)
  • Received : 2021.08.27
  • Accepted : 2021.11.01
  • Published : 2021.12.25
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Abstract

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

Keywords

Acknowledgement

MJ was funded, in part, by the Doctoral Dissertation Fellowship of the Department of Mathematics at the University at Buffalo. JHJ has been supported by Samsung Science & Technology Foundation under grant number SSTF-BA1802-02.

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