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}
}
  • A. Cerri, C. Landi
  • Published 1 April 2016
  • Mathematics
  • Foundations of Computational Mathematics
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… 
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12 Citations
Background Citations
7
Methods Citations
2

12 Citations

Morse-based Fibering of the Persistence Rank Invariant
TLDR
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
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
TLDR
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
TLDR
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
Persistent Homology as Stopping-Criterion for Natural Neighbor Interpolation
TLDR
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
Persistent Homology as Stopping-Criterion for Voronoi Interpolation
TLDR
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
  • C. Landi
  • Mathematics, Computer Science
    ArXiv
  • 2014
TLDR
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
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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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References

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TLDR
The persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense and a method to visualize topological features of a shape via persistence spaces is presented.
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TLDR
This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.
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TLDR
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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…
Betti numbers in multidimensional persistent homology are stable functions
Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of…
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