Edit Distance and Persistence Diagrams Over Lattices
Abstract
We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice and the output is a persistence diagram defined as the Möbius inversion of its birthdeath function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are $1$Lipschitz making our pipeline stable. Our constructions generalize the classical persistence diagram and, in this setting, the bottleneck distance is strongly equivalent to the edit distance.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.07337
 arXiv:
 arXiv:2010.07337
 Bibcode:
 2020arXiv201007337M
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Computational Geometry
 EPrint:
 Accepted to SIAM Journal on Applied Geometry and Algebra. This is the final accepted version of the paper