Betti numbers in multidimensional persistent homology are stable functions

By Andrea Cerri, B. D. Fabio, Massimo Ferri, Patrizio Frosini, C. Landi. 2013

162 citations recorded by the source corpus.

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 scalar‐valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, that is, the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector‐valued filtering functions, we can consider the multidimensional analog of persistent Betti numbers. Varying the lower level sets, we obtain that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non‐negative integers. In this paper, we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector‐valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector‐valued filtering functions and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalar‐valued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers, we obtain a lower bound for the natural pseudo‐distance. Copyright © 2013 John Wiley & Sons, Ltd.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3f844c81535529cfeb56c823cbfa14b41a0d1873

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Andreas Holzinger
2. Chengyuan Wu
3. Jie Wu
4. Heimo Müller
5. S. Ren

In Mathematical Physics

1. Chengyuan Wu
2. Jie Wu
3. S. Ren

Citations in the corpus

• Weighted persistent homology (2017, background): “In [5], the authors Cerri, Fabio, Ferri, Frosini and Landi show that Betti numbers in multidimensional persistent homology are stable functions, in the sense that small changes of the vector-valued filtering functions imply only small changes of persistent Betti numbers functions.”
• Towards the Augmented Pathologist: Challenges of Explainable-AI in Digital Pathology (2017, background): “Also interesting for our work are simplicial complexes (”simplicials”) which are spaces described in a very particular way, the basis of which is homology (Cerri et al., 2013).”