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Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces

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Abstract

We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces and illustrate their use in hypothesis testing and providing confidence intervals for topological data analysis.

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Acknowledgments

The authors would like to thank Gunnar Carlsson and Michael Lesnick for useful comments, Rachel Ward for comments on a prior draft, and Olena Blumberg for help with background and for assistance with the analysis of the tightness of the main theorem. We would also like to thank the Institute for Mathematics and Its Applications for hospitality while revising this paper. The authors were supported in part by Defense Advanced Research Projects Agency (DARPA) Young Faculty Award N66001-10-1-4043.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Andrew J. Blumberg, Itamar Gal & Matthew Pancia

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Michael A. Mandell

Authors
  1. Andrew J. Blumberg
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  2. Itamar Gal
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  3. Michael A. Mandell
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Correspondence to Andrew J. Blumberg.

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Communicated by Gunnar Carlsson.

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Blumberg, A.J., Gal, I., Mandell, M.A. et al. Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces. Found Comput Math 14, 745–789 (2014). https://doi.org/10.1007/s10208-014-9201-4

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