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.