Geometric Metrics for Topological Representations

Anirudh Som (), Karthikeyan Natesan Ramamurthy () and Pavan Turaga ()
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Anirudh Som: School of Arts, Media and Engineering, Arizona State University, School of Electrical, Computer and Energy Engineering
Karthikeyan Natesan Ramamurthy: IBM Research
Pavan Turaga: School of Arts, Media and Engineering, Arizona State University, School of Electrical, Computer and Energy Engineering

Chapter Chapter 15 in Handbook of Variational Methods for Nonlinear Geometric Data, 2020, pp 415-441 from Springer

Abstract: 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.

Date: 2020
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DOI: 10.1007/978-3-030-31351-7_15

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