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arXiv · 1603.07406

Higher interpolation and extension for persistence modules

Peter Bubenik·Vin de Silva·Vidit Nanda
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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 \v{C}ech complexes built within the space of persistence modules.

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Peter Bubenik, Vin de Silva, Vidit Nanda. 2016-03-24. Higher interpolation and extension for persistence modules. https://doi.org/10.1137/16m1100472

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