How Small Can Faithful Sets Be? Ordering Topological Descriptors
Provides a theoretical foundation for comparing topological descriptors, aiding practitioners in choosing descriptor types for shape reconstruction and comparison.
The paper establishes a framework to quantitatively compare topological descriptors and partially orders six common types, showing that verbose descriptors require smaller faithful sets than concise ones.
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.