# 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} }

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…

## 18 Citations

Topological spaces of persistence modules and their properties

- MathematicsJ. Appl. Comput. Topol.
- 2018

This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties.

Interleaving and Gromov-Hausdorff distance

- Mathematics
- 2017

One of the central notions to emerge from the study of persistent homology is that of interleaving distance. It has found recent applications in symplectic and contact geometry, sheaf theory,…

Local Cohomology and Stratification

- MathematicsFound. Comput. Math.
- 2020

An algorithm to recover the canonical stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum.

Homotopy, homology, and persistent homology using closure spaces and filtered closure spaces

- Mathematics
- 2021

. We develop persistent homology in the setting of ﬁltered (ˇCech) closure spaces. Examples of ﬁltered closure spaces include ﬁltered topological spaces, metric spaces, weighted graphs, and weighted…

A T ] 2 5 O ct 2 01 8 TOPOLOGICAL SPACES OF PERSISTENCE MODULES AND THEIR PROPERTIES

- Mathematics
- 2018

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We…

Generalized persistence diagrams

- MathematicsJ. Appl. Comput. Topol.
- 2018

The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer is generalized to the setting of constructible persistence modules valued in a symmetric monoidal category and a second type of persistence diagram is defined, which enjoys a stronger stability theorem.

Computational Complexity of the Interleaving Distance

- Mathematics, Computer ScienceSoCG
- 2018

It is shown that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces, and it is obtained that the isomorphism problem for Reeb graphs is graph isomorphicism complete.

Topological Methods in Data Analysis

- Mathematics
- 2017

I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete…

Positivity of Multiparameter Persistence Diagrams and Bottleneck Stability

- Mathematics
- 2019

Persistent homology studies the birth and death of cycles in a parameterized family of spaces. In this paper, we study the birth and death of cycles in a multifiltration of a chain complex with the…

Connectedness and Lusternik-Schnirelmann categories of the spaces of persistence modules

- Mathematics
- 2020

The classes of various interval decomposable persistence modules in literature were analyzed, the sets were determined, and some topological characteristics that these sets gained through…

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