Edit Distance and Persistence Diagrams over Lattices
Alexander McCleary, Amit Patel
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 birth-death 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.
Topics & Concepts
Persistent homologyTopological data analysisMathematicsBottleneckFunctorEdit distanceMetric spaceLattice (music)Pure mathematicsDiscrete mathematicsTopology (electrical circuits)CombinatoricsComputer scienceAlgorithmAcousticsPhysicsEmbedded systemTopological and Geometric Data AnalysisHomotopy and Cohomology in Algebraic TopologyAdvanced Neuroimaging Techniques and Applications