Ordering Topological Descriptors
CoRR(2024)
摘要
Recent developments in shape reconstruction and comparison call for the use
of many different types of topological descriptors (persistence diagrams, Euler
characteristic functions, etc.). We establish a framework that allows for
quantitative comparisons of topological descriptor types and therefore may be
used as a tool in more rigorously justifying choices made in applications. We
then use this framework to partially order a set of six common topological
descriptor types. In particular, the resulting poset gives insight into the
advantages of using verbose rather than concise topological descriptors. We
then provide lower bounds on the size of sets of descriptors that are complete
discrete invariants of simplicial complexes, both tight and worst case. This
work sets up a rigorous theory that allows for future comparisons and analysis
of topological descriptor types.
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