Cerri, Andrea ;
Di Fabio, Barbara ;
Ferri, Massimo ;
Frosini, Patrizio ;
Landi, Claudia
(2009)
Multidimensional persistent homology is stable.
p. 14.
DOI
10.6092/unibo/amsacta/2603.
Full text available as:
Abstract
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued
functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove
that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.
Abstract
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued
functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove
that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.
Document type
Monograph
(Technical Report)
Creators
Keywords
Multidimensional persistence, persistence diagram, size function, Cech homology, foliation
Subjects
DOI
Deposit date
03 Aug 2009
Last modified
16 May 2011 12:10
URI
Other metadata
Document type
Monograph
(Technical Report)
Creators
Keywords
Multidimensional persistence, persistence diagram, size function, Cech homology, foliation
Subjects
DOI
Deposit date
03 Aug 2009
Last modified
16 May 2011 12:10
URI
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Multidimensional persistent homology is stable. (deposited 03 Aug 2009)
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