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Foundations of Computational Mathematics

, Volume 17, Issue 1, pp 1–33 | Cite as

Persistence Barcodes Versus Kolmogorov Signatures: Detecting Modes of One-Dimensional Signals

  • Ulrich Bauer
  • Axel Munk
  • Hannes Sieling
  • Max Wardetzky
Article

Abstract

We investigate the problem of estimating the number of modes (i.e., local maxima)—a well-known question in statistical inference—and we show how to do so without presmoothing the data. To this end, we modify the ideas of persistence barcodes by first relating persistence values in dimension one to distances (with respect to the supremum norm) to the sets of functions with a given number of modes, and subsequently working with norms different from the supremum norm. As a particular case, we investigate the Kolmogorov norm. We argue that this modification has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples.

Keywords

Persistent homology Mode hunting Exponential deviation bound Partial sum process Taut strings 

Mathematics Subject Classification

Primary 62G05 62G20 68U05 Secondary 62H12 57R70 58E05 

Notes

Acknowledgments

We would like to thank the anonymous reviewers for their very helpful suggestions for revising our manuscript and Carola Schoenlieb for inspiring discussions.

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Copyright information

© SFoCM 2015

Authors and Affiliations

  • Ulrich Bauer
    • 1
  • Axel Munk
    • 2
    • 3
  • Hannes Sieling
    • 2
  • Max Wardetzky
    • 4
  1. 1.Technische Universität München (TUM)MunichGermany
  2. 2.Institute for Mathematical StochasticsUniversity of GöttingenGöttingenGermany
  3. 3.Max Planck Institute for Biophysical ChemistryGöttingenGermany
  4. 4.Institute of Numerical and Applied MathematicsUniversity of GöttingenGöttingenGermany

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