# One-dimensional reduction of multidimensional persistent homology

@inproceedings{Cagliari2007OnedimensionalRO, title={One-dimensional reduction of multidimensional persistent homology}, author={Francesca Cagliari and Barbara Di Fabio and Massimo Ferri}, year={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 stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.

## 53 Citations

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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.

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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.

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A global method for reducing size graphs is presented, together with a theorem stating that size graphs reduced in such a way preserve all the information in terms of multidimensional size functions.

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In this paper, we generalize the embedded homology in [1] for hypergraphs and study the relative embedded homology for hypergraph pairs. We study the topology for sub-hypergraphs. Using the relative…

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