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} }
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…
12 Citations
Morse-based Fibering of the Persistence Rank Invariant
- MathematicsArXiv
- 2020
It is shown how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field.
A T ] 3 0 N ov 2 02 0 Morse-based Fibering of the Persistence Rank Invariant Asilata
- Mathematics
- 2020
Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available…
The Coherent Matching Distance in 2D Persistent Homology
- Computer ScienceCTIC
- 2016
This paper introduces a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations the authors take into account, and proves that the coherent 2D matching distance is well-defined and stable.
On the geometrical properties of the coherent matching distance in 2D persistent homology
- MathematicsJ. Appl. Comput. Topol.
- 2019
A new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way is studied, including its stability.
Rigorous cubical approximation and persistent homology of continuous functions
- Mathematics, Computer ScienceComput. Math. Appl.
- 2018
Persistent Homology as Stopping-Criterion for Voronoi Interpolation
- Computer ScienceIWCIA
- 2020
The Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration to capture the changing topology of the data.
The rank invariant stability via interleavings
- Mathematics, Computer ScienceArXiv
- 2014
A lower bound for the interleaving distance on persistence modules is given in terms of matching distance of rank invariants, and the internal stability of the rank invariant is proved in Terms of interleavings.
Combinatorial Image Analysis: 20th International Workshop, IWCIA 2020, Novi Sad, Serbia, July 16–18, 2020, Proceedings
- MathematicsIWCIA
- 2020
It is proved in this paper that, first, self-dual Euler wellcomposedness is equivalent to digital well- Composedness in dimension 2 and 3, and second, in dimension 4,Self-duals well-Composedness impliesdigital well-composeness, though the converse is not true.
3D Artifacts Similarity Based on the Concurrent Evaluation of Heterogeneous Properties
- Computer ScienceJOCCH
- 2015
A shape analysis and comparison pipeline specifically targeted to the similarity assessment of real-world 3D artifacts and the potential of the approach is high because any property representable as real- or vector- valued functions can be easily added in the framework.
Computing multiparameter persistent homology through a discrete Morse-based approach
- Computer ScienceComput. Geom.
- 2020
References
SHOWING 1-10 OF 13 REFERENCES
The Persistence Space in Multidimensional Persistent Homology
- MathematicsDGCI
- 2013
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.
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.
Necessary conditions for discontinuities of multidimensional persistent Betti numbers
- Mathematics
- 2015
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…
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…
Comparison of persistent homologies for vector functions: From continuous to discrete and back
- MathematicsComput. Math. Appl.
- 2013
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…
Size homotopy groups for computation of natural size distances
- Mathematics
- 1999
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…
Size Functions and Formal Series
- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2001
It is proved that every size function can be represented as a set of points and lines in the real plane, with multiplicities, which allows for an algebraic approach to size functions and the construction of new pseudo-distances between size functions for comparing shapes.