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 generalize the concept of persistence diagram in this sense. The main result is its stability under function perturbations: Any change in vector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.

Hausdorff Stability of Persistence Spaces / Cerri, Andrea; Landi, Claudia. - In: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS. - ISSN 1615-3375. - STAMPA. - 16:(2016), pp. 343-367. [10.1007/s10208-015-9244-1]

Hausdorff Stability of Persistence Spaces

LANDI, Claudia
2016

Abstract

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 generalize the concept of persistence diagram in this sense. The main result is its stability under function perturbations: Any change in vector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.
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343
367
Hausdorff Stability of Persistence Spaces / Cerri, Andrea; Landi, Claudia. - In: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS. - ISSN 1615-3375. - STAMPA. - 16:(2016), pp. 343-367. [10.1007/s10208-015-9244-1]
Cerri, Andrea; Landi, Claudia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1065326
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