# 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} }

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…

## 4 Citations

Erosion distance for generalized persistence modules

- Mathematics
- 2017

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

- MathematicsFound. Comput. Math.
- 2015

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

- Computer Science
- 2013

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

- MathematicsFound. Comput. Math.
- 2015

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

- Mathematics
- 2009

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

- MathematicsSCG '07
- 2007

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

- Mathematics, Computer ScienceSCG '09
- 2009

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

- Computer ScienceArXiv
- 2008

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

- Mathematics
- 2007

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

- Mathematics, Computer ScienceSGP '04
- 2004

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

- Mathematics
- 2009

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

- Mathematics

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

- MathematicsSCG
- 2005

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

- MathematicsAppl. Math. Lett.
- 2011