Finiteness of rank invariants of multidimensional persistent homology groups

@article{Cagliari2011FinitenessOR,
  title={Finiteness of rank invariants of multidimensional persistent homology groups},
  author={Francesca Cagliari and Claudia Landi},
  journal={ArXiv},
  year={2011},
  volume={abs/1001.0358}
}
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26 Citations
Background Citations
15
Methods Citations
1
Results Citations
1

26 Citations

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Citation Type

Parametrized homology via zigzag persistence
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological…
The Persistence Space in Multidimensional Persistent Homology
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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.
Persistent Homology and the Upper Box Dimension
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A fractal dimension for a metric space based on the persistent homology of subsets of that space is introduced and hypotheses under which this dimension is comparable to the upper box dimension are exhibited.
Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
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A pseudo-distance dT is introduced that represents a possible solution to the present lack of a stable method to compare persistent homology groups with torsion, and the main theorem proves the stability of the new pseudo- distance with respect to the change of the filtering function.
The Persistent Homology of Random Geometric Complexes on Fractals
We study the asymptotic behavior of the persistent homology of i.i.d. samples from a d-Ahlfors regular measure on a metric space — one that satisfies uniform bounds of the form 1 c r ≤ μ (Br (x)) ≤ c…
Geometry in the space of persistence modules
TLDR
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.
Uniqueness of models in persistent homology: the case of curves
We consider generic curves in , i.e. generic C1 functions . We analyze these curves through the persistent homology groups of a filtration induced on S1 by f. In particular, we consider the question…
Comparison of persistent homologies for vector functions: From continuous to discrete and back
Sketches of a platypus: persistent homology and its algebraic foundations
TLDR
The various choices in use, and what they allow us to prove are examined, and the inherent differences between the choices people use are discussed, and potential directions of research are speculated on.
G‐invariant persistent homology
TLDR
The stability of G-invariant persistent homology with respect to the natural pseudo-distance d_G is proved, and it is shown how this idea could be used in applications concerning shape comparison.
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