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Quantitative Homotopy Theory in Topological Data Analysis

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Abstract

This paper lays the foundations of an approach to applying Gromov’s ideas on quantitative topology to topological data analysis. We introduce the “contiguity complex”, a simplicial complex of maps between simplicial complexes defined in terms of the combinatorial notion of contiguity. We generalize the Simplicial Approximation Theorem to show that the contiguity complex approximates the homotopy type of the mapping space as we subdivide the domain. We describe algorithms for approximating the rate of growth of the components of the contiguity complex under subdivision of the domain; this procedure allows us to computationally distinguish spaces with isomorphic homology but different homotopy types.

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Acknowledgements

This research supported in part by DARPA YFA 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

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

    Michael A. Mandell

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

Additional information

Communicated by Gunnar Carlsson.

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Blumberg, A.J., Mandell, M.A. Quantitative Homotopy Theory in Topological Data Analysis. Found Comput Math 13, 885–911 (2013). https://doi.org/10.1007/s10208-013-9177-5

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  • DOI: https://doi.org/10.1007/s10208-013-9177-5

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