Higher interpolation and extension for persistence modules

Abstract

The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. Our main construction gives an isometric embedding of a metric space into the metric space of persistence modules with values in the spacetime of this metric space. As a consequence of such "higher interpolation", it becomes possible to compare Vietoris-Rips and Čech complexes built within the space of persistence modules.

12 Pages, 2 Figures

References in corpus (3)

• Analysis of Kolmogorov Flow and Rayleigh-Bénard Convection using Persistent Homology
• The structure and stability of persistence modules
• Topological Signals of Singularities in Ricci Flow

Cited by in corpus (6)

• Generalized Persistence Diagrams
• A Survey of Vectorization Methods in Topological Data Analysis
• Bottleneck Stability for Generalized Persistence Diagrams
• Topological spaces of persistence modules and their properties
• Homotopy, homology, and persistent homology using closure spaces
• Positivity of Multiparameter Persistence Diagrams and Bottleneck Stability