# Hausdorff Stability of Persistence Spaces

@article{Cerri2016HausdorffSO, title={Hausdorff Stability of Persistence Spaces}, author={Andrea Cerri and Claudia Landi}, journal={Foundations of Computational Mathematics}, year={2016}, volume={16}, pages={343-367} }

Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to…

## 12 Citations

Morse-based Fibering of the Persistence Rank Invariant

- MathematicsArXiv
- 2020

It is shown how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field.

A T ] 3 0 N ov 2 02 0 Morse-based Fibering of the Persistence Rank Invariant Asilata

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- 2020

Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available…

The Coherent Matching Distance in 2D Persistent Homology

- Computer ScienceCTIC
- 2016

This paper introduces a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations the authors take into account, and proves that the coherent 2D matching distance is well-defined and stable.

On the geometrical properties of the coherent matching distance in 2D persistent homology

- MathematicsJ. Appl. Comput. Topol.
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A new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way is studied, including its stability.

Rigorous cubical approximation and persistent homology of continuous functions

- Mathematics, Computer ScienceComput. Math. Appl.
- 2018

Persistent Homology as Stopping-Criterion for Natural Neighbor Interpolation

- Computer ScienceArXiv
- 2019

In this study the method of natural neighbours is used to interpolate data that has been drawn from a topological space with higher homology groups on its filtration to capture the changing topology of the data.

Computing multiparameter persistent homology through a discrete Morse-based approach

- Computer ScienceComput. Geom.
- 2020

Persistent Homology as Stopping-Criterion for Voronoi Interpolation

- Computer ScienceIWCIA
- 2020

The Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration to capture the changing topology of the data.

The rank invariant stability via interleavings

- Mathematics, Computer ScienceArXiv
- 2014

A lower bound for the interleaving distance on persistence modules is given in terms of matching distance of rank invariants, and the internal stability of the rank invariant is proved in Terms of interleavings.

Combinatorial Image Analysis: 20th International Workshop, IWCIA 2020, Novi Sad, Serbia, July 16–18, 2020, Proceedings

- MathematicsIWCIA
- 2020

It is proved in this paper that, first, self-dual Euler wellcomposedness is equivalent to digital well- Composedness in dimension 2 and 3, and second, in dimension 4,Self-duals well-Composedness impliesdigital well-composeness, though the converse is not true.

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