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Geometric Metrics for Topological Representations

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Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

In this chapter, we present an overview of recent techniques from the emerging area of topological data analysis (TDA), with a focus on machine-learning applications. TDA methods are concerned with measuring shape-related properties of point-clouds and functions, in a manner that is invariant to topological transformations. With a careful design of topological descriptors, these methods can result in a variety of limited, yet practically useful, invariant representations. The generality of this approach results in a flexible design choice for practitioners interested in developing invariant representations from diverse data sources such as image, shapes, and time-series data. We present a survey of topological representations and metrics on those representations, discuss their relative pros and cons, and illustrate their impact on a few application areas of recent interest.

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Acknowledgements

This work was supported in part by NSF CAREER grant 1452163 and ARO grant number W911NF-17-1-0293 during the preparation of this chapter.

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Authors and Affiliations

  1. School of Electrical, Computer and Energy Engineering, School of Arts, Media and Engineering, Arizona State University, Tempe, AZ, USA

    Anirudh Som & Pavan Turaga

  2. IBM Research, Yorktown Heights, NY, USA

    Karthikeyan Natesan Ramamurthy

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  1. Anirudh Som
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Correspondence to Anirudh Som .

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  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Som, A., Ramamurthy, K.N., Turaga, P. (2020). Geometric Metrics for Topological Representations. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_15

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