# Multidimensional persistent homology is stable

@article{Cerri2009MultidimensionalPH, title={Multidimensional persistent homology is stable}, author={Andrea Cerri and Barbara Di Fabio and Massimo Ferri and Patrizio Frosini and Claudia Landi}, journal={arXiv: Algebraic Topology}, year={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 analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with re- spect to function perturbations. More precisely, we construct a distance be- tween rank invariants such that small changes of the function imply only small changes of the rank invariant. This result…

## 16 Citations

### Invariance properties of the multidimensional matching distance in Persistent Topology and Homology

- MathematicsArXiv
- 2010

This paper shows that the multidimensional matching distance is actually invariant with respect to such a choice, and formally depends on a subset of $\R^n\times-valued filtering functions inducing a parameterization of these half-planes.

### Finiteness of rank invariants of multidimensional persistent homology groups

- MathematicsAppl. Math. Lett.
- 2011

### Stable Comparison of Multidimensional Persistent Homology Groups with Torsion

- MathematicsArXiv
- 2010

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.

### Stability of multidimensional persistent homology with respect to domain perturbations

- Computer ScienceArXiv
- 2010

It is shown that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance between sets.

### Biparametric persistence for smooth filtrations

- Mathematics
- 2021

The goal of this note is to define biparametric persistence diagrams for smooth generic mappings h = (f,g) : M → V ∼ = R for smooth compact manifoldM. Existing approaches to multivariate persistence…

### Estimating Multidimensional Persistent Homology Through a Finite Sampling

- MathematicsInt. J. Comput. Geom. Appl.
- 2015

It is shown that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of X from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain.

### A new approximation Algorithm for the Matching Distance in Multidimensional Persistence

- Computer Science
- 2011

This paper proposes a new computational framework to deal with the multidimensional matching distance, by proving some new theoretical results and using them to formulate an algorithm for computing such a distance up to an arbitrary threshold error.

### Persistence and Computation of the Cup Product

- Mathematics
- 2010

In this paper we discuss the computation of the cup product for the cohomology of a finite simplicial complex over a field. Working with the theory of persistent (co)homology, we study an algebraic…

### A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions

- MathematicsFound. Comput. Math.
- 2011

It is shown that persistence diagrams are able to recognize an occluded shape by showing a common subset of points and a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms is obtained.

### Sketches of a platypus: persistent homology and its algebraic foundations

- MathematicsArXiv
- 2012

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.

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