  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.
