# Higher Interpolation and Extension for Persistence Modules

@article{Bubenik2017HigherIA,
title={Higher Interpolation and Extension for Persistence Modules},
author={Peter Bubenik and Vin de Silva and Vidit Nanda},
journal={SIAM J. Appl. Algebra Geom.},
year={2017},
volume={1},
pages={272-284}
}
• Published 24 March 2016
• Mathematics
• SIAM J. Appl. Algebra Geom.
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…
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## References

SHOWING 1-10 OF 33 REFERENCES
Proximity of persistence modules and their diagrams
• Mathematics, Computer Science
SCG '09
• 2009
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
Geometry in the space of persistence modules
• Mathematics
SoCG '13
• 2013
It is shown that the relationship between the Cech and Rips complexes is governed by certain `coherence' conditions on the corresponding families of interleavings or matchings in the spaces of persistence modules and diagrams.
Metrics for Generalized Persistence Modules
• Mathematics
Found. Comput. Math.
• 2015
This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems.
The Structure and Stability of Persistence Modules
• Mathematics
Springer Briefs in Mathematics
• 2016
This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such
Stability of persistence diagrams
• Mathematics
SCG
• 2005
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram
Topological Signatures of Singularities in Simplicial Ricci Flow
• Mathematics
• 2015
We apply the methods of persistent homology to a selection of two and three--dimensional geometries evolved by simplicial Ricci flow. To implement persistent homology, we construct a triangular mesh
The observable structure of persistence modules
• Mathematics
• 2014
In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence
Topological Signals of Singularities in Ricci Flow
• Mathematics
Axioms
• 2017
The results obtained suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formations (neckpinch) under Ricci flow.
Categorification of Persistent Homology
• Mathematics
Discret. Comput. Geom.
• 2014
This work redevelops persistent homology (topological persistence) from a categorical point of view and gives a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category and shows that if the target category is abelian, so is this category of interleavments.
Statistical topological data analysis using persistence landscapes
A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined.