Geometry in the space of persistence modules

2013/06/17 by Vin de Silva, Vidit Nanda · 5 citations
Computer Science · Medicine · Mathematics · #Topological and Geometric Data Analysis #Advanced Neuroimaging Techniques and Applications #Topological data analysis #Persistence (discontinuity) #Interleaving #Simplicial complex #Point cloud #Persistent homology #Metric space #Diagram #Metric (unit) #Mathematics #Topology (electrical circuits) #Topological space #Pure mathematics #Computer science #Algebra over a field #Combinatorics #Algorithm #Artificial intelligence

paper · doi:10.1145/2462356.2462402

openalex publication_date 2013/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

Topological persistence is, by now, an established paradigm for constructing robust topological invariants from point cloud data: the data are converted into a filtered simplicial complex, the complex gives rise to a persistence module, and the module is described by a persistence diagram. In this paper, we study the geometry of the spaces of persistence modules and diagrams, with special attention to Cech and Rips complexes. The metric structures are determined in terms of interleaving maps between modules and matchings between diagrams. We show that the relationship between the Cech and Rips complexes is governed by certain `coherence' conditions on the corresponding families of interleavings or matchings.

Citations

• Topology and data
• Morse Theory for Filtrations and Efficient Computation of Persistent Homology
• Stability of Persistence Diagrams
• Probability measures on the space of persistence diagrams
• Decomposition of graded modules
• The Theory of the Interleaving Distance on Multidimensional Persistence Modules
• Barcodes: The persistent topology of data
• Computing Persistent Homology
• Finiteness of rank invariants of multidimensional persistent homology groups
• Topological Persistence and Simplification
• Persistent homology—a survey

Cited by

• Topological data analysis [wikipedia]