# 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}
}
• Published 23 February 2007
• Mathematics
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.
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Topological Persistence and Simplification
• Economics
Discret. Comput. Geom.
• 2002
Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
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
A distance for similarity classes of submanifolds of a Euclidean space
• P. Frosini
• Mathematics, Computer Science
Bulletin of the Australian Mathematical Society
• 1990
A distance is defined on the quotient of the set of submanifolds of a Euclidean space, with respect to similarity. It is then related to a previously defined function which captures the metric