Why Should Q=P in the Wasserstein Distance Between Persistence Diagrams?

Explore a comprehensive lecture on the importance of using q=p in the Wasserstein distance between persistence diagrams. Delve into five key reasons supporting this choice, including improved formula simplicity, better local geometry reflection, natural stability results, algebraic version definition for persistence modules, and easier computation of central tendencies. Examine various examples, including height functions and point clouds, to illustrate the concepts. Investigate topics such as matchings between diagrams, bottleneck distance, sublevel sets of functions on simplicial complexes, and interleaving distance. Learn about the p-norm of a persistence module, morphisms between persistence diagrams, and the construction of spans from matchings. Analyze the mean as a minimizer of the sum of distances squared and explore candidates for mean and median calculations. Gain insights into Lipschitz stability corollaries and understand the case for adopting q=p in the Wasserstein distance formula.

Intro
Matchings between diagrams
Bottleneck distance distance
The main contenders
Coordinates have separate meanings
An example with height functions
An example with point clouds
Recall: Sublevel sets of functions on simplicial complexes
Local Stability for functions on simplicial complexes
Interleaving distance
The p-norm of a persistence module
Morphisms between persistence diagrams
Example with persistence modules of a single interval
Constructing a span from a matching
Spans for the bottleneck distance - matching the diagonal
Mean as minimiser of sum of distances squared
Candidates for the Mean
Candidates for the Median
Median of a selection - q=p=1
A case for change - replace
Lipschitz stability corollaries