Stable Comparison of Multidimensional Persistent Homology Groups with Torsion

@article{Frosini2010StableCO,
  title={Stable Comparison of Multidimensional Persistent Homology Groups with Torsion},
  author={Patrizio Frosini},
  journal={Acta Applicandae Mathematicae},
  year={2010},
  volume={124},
  pages={43-54}
}
  • P. Frosini
  • Published 19 December 2010
  • Mathematics
  • Acta Applicandae Mathematicae
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance dT that represents a possible solution to this problem. Indeed, dT is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the… 
Erosion distance for generalized persistence modules
The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with
Metrics for Generalized Persistence Modules
TLDR
This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems.
1 Theory of Interleavings and Interleaving Distances On Mutidimensional Persistence Modules
TLDR
It is believed that a more fully developed methodology for TDA would greatly broaden the utility and appeal of these tools to statisticians and scientists, and would thus hasten the discovery of applications of TDA to the sciences.
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
TLDR
The theory of multidimensional interleavings is developed, with a view toward applications to topological data analysis, and it is shown that when the authors define their persistence modules over a prime field, d_\mathrm{I}$$dI satisfies a universality property.

References

SHOWING 1-10 OF 32 REFERENCES
Multidimensional persistent homology is stable
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional
The theory of multidimensional persistence
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.
Proximity of persistence modules and their diagrams
TLDR
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.
Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions
TLDR
The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance, and it is proved that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.
One-dimensional reduction of multidimensional persistent homology
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
Persistence barcodes for shapes
TLDR
This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.
Natural pseudo-distances between closed curves
Abstract Let us consider two closed curves ℳ, of class C 1 and two functions of class C 1, called measuring functions. The natural pseudo-distance d between the pairs (ℳ, φ), (, ψ) is defined as the
Persistent Homology — a Survey
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical
Stability of persistence diagrams
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
Finiteness of rank invariants of multidimensional persistent homology groups
...
...