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Hausdorff Stability of Persistence Spaces

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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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Acknowledgments

Work carried out within the activity of ARCES “E. De Castro”, University of Bologna, under the auspices of INdAM-GNSAGA. Andrea Cerri was partially supported by the CNR research activity ICT.P10.009.

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Authors and Affiliations

  1. Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, Consiglio Nazionale Delle Ricerche, Via de Marini 6, 16149, Genova, Italy

    Andrea Cerri

  2. Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, Via Amendola 2, Pad. Morselli, 42100, Reggio Emilia, Italy

    Claudia Landi

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  1. Andrea Cerri
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  2. Claudia Landi
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Correspondence to Claudia Landi.

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Communicated by Gunnar Carlsson.

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Cerri, A., Landi, C. Hausdorff Stability of Persistence Spaces. Found Comput Math 16, 343–367 (2016). https://doi.org/10.1007/s10208-015-9244-1

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