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Geometric metrics for topological representations. (English) Zbl 1512.62108

Grohs, Philipp (ed.) et al., Handbook of variational methods for nonlinear geometric data. Cham: Springer. 415-441 (2020).
Summary: 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.
For the entire collection see [Zbl 07115003].

MSC:

62R40 Topological data analysis
54H30 Applications of general topology to computer science (e.g., digital topology, image processing)
57Z25 Relations of manifolds and cell complexes with computer and data science
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