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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References

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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.
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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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This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
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Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account
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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
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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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A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a
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For every manifold M endowed with a structure described by a function from M to the vector space R k , a parametric family of groups, called size homotopy groups, is introduced and studied. Some
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