# Proximity of persistence modules and their diagrams

@inproceedings{Chazal2009ProximityOP, title={Proximity of persistence modules and their diagrams}, author={Fr{\'e}d{\'e}ric Chazal and David Cohen-Steiner and Marc Glisse and Leonidas J. Guibas and Steve Oudot}, booktitle={SCG '09}, year={2009} }

Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces. In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level…

## 395 Citations

Higher Interpolation and Extension for Persistence Modules

- MathematicsSIAM J. Appl. Algebra Geom.
- 2017

A coherence criterion is isolated which guarantees the extensibility of non-expansive maps into this space of persistence modules across embeddings of the domain to larger ambient metric spaces, allowing Kan extensions to provide the desired extensions.

Metric Geometry of Spaces of Persistence Diagrams

- Mathematics
- 2021

. Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence…

Geometry in the space of persistence modules

- MathematicsSoCG '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.

Rank-based persistence

- MathematicsArXiv
- 2019

This work gives axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions and proves the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories.

Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stability

- MathematicsRevista Matemática Complutense
- 2018

A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply…

Hausdorff Stability of Persistence Spaces

- MathematicsFound. Comput. Math.
- 2016

The persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense and the main result is its stability under function perturbations.

Correspondence Modules and Persistence Sheaves: A Unifying Framework for One-Parameter Persistent Homology

- Mathematics
- 2020

We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are…

Stability of persistence spaces of vector-valued continuous functions

- MathematicsArXiv
- 2013

The persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram and its main result is its stability under function perturbations: any change invector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.

Algebraic Stability of Zigzag Persistence Modules

- MathematicsAlgebraic & Geometric Topology
- 2018

This paper functorially extends each zigzag persistence module to a two-dimensional persistence module, and establishes an algebraic stability theorem for these extensions, which yields a stability result for free two- dimensional persistence modules.

The Persistent Homotopy Type Distance

- MathematicsHomology, Homotopy and Applications
- 2019

This work proves that dHT extends the L-infty distance and dNP in two ways and shows that, appropriately restricting the category of objects to which dHT applies, it can be made to coincide with the other two distances.

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