How Small Can Faithful Sets Be? Ordering Topological Descriptors

How Small Can Faithful Sets Be? Ordering Topological Descriptors

Recent developments in shape reconstruction and comparison call for the use of many different (topological) descriptor types, such as persistence diagrams and Euler characteristic functions. We establish a framework to quantitatively compare the strength of different descriptor types, setting up a t...

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Veröffentlicht in:arXiv.org 2024-07
Hauptverfasser: Fasy, Brittany Terese, Millman, David L, Schenfisch, Anna
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Characteristic functions
Lower bounds
Set theory
Topology
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description Recent developments in shape reconstruction and comparison call for the use of many different (topological) descriptor types, such as persistence diagrams and Euler characteristic functions. We establish a framework to quantitatively compare the strength of different descriptor types, setting up a theory that allows for future comparisons and analysis of descriptor types and that can inform choices made in applications. We use this framework to partially order a set of six common descriptor types. We then give lower bounds on the size of sets of descriptors that uniquely correspond to simplicial complexes, giving insight into the advantages of using verbose rather than concise topological descriptors.
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subjects Characteristic functions
Lower bounds
Set theory
Topology
title How Small Can Faithful Sets Be? Ordering Topological Descriptors
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