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Natural pseudodistance

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In size theory, the natural pseudodistance between two size pairs , is the value , where varies in the set of all homeomorphisms from the manifold to the manifold and is the supremum norm. If and are not homeomorphic, then the natural pseudodistance is defined to be . It is usually assumed that , are closed manifolds and the measuring functions are . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from to .

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function takes values in .[1] When , the group of all homeomorphisms of can be replaced in the definition of natural pseudodistance by a subgroup of , so obtaining the concept of natural pseudodistance with respect to the group .[2][3] Lower bounds and approximations of the natural pseudodistance with respect to the group can be obtained both by means of -invariant persistent homology[4] and by combining classical persistent homology with the use of G-equivariant non-expansive operators.[2][3]

Main properties

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It can be proved [5] that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions (possibly, of the same measuring function) divided by a suitable positive integer . If and are surfaces, the number can be assumed to be , or .[6] If and are curves, the number can be assumed to be or .[7] If an optimal homeomorphism exists (i.e., ), then can be assumed to be .[5] The research concerning optimal homeomorphisms is still at its very beginning .[8][9]


See also

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References

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  1. ^ Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455-464, 1999.
  2. ^ a b Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, 55(2):373-409, 2016.
  3. ^ a b Mattia G. Bergomi, Patrizio Frosini, Daniela Giorgi, Nicola Quercioli, Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning, Nature Machine Intelligence, (2 September 2019). DOI: 10.1038/s42256-019-0087-3 Full-text access to a view-only version of this paper is available at the link https://rdcu.be/bP6HV .
  4. ^ Patrizio Frosini, G-invariant persistent homology, Mathematical Methods in the Applied Sciences, 38(6):1190-1199, 2015.
  5. ^ a b Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.
  6. ^ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.
  7. ^ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.
  8. ^ Andrea Cerri, Barbara Di Fabio, On certain optimal diffeomorphisms between closed curves, Forum Mathematicum, 26(6):1611-1628, 2014.
  9. ^ Alessandro De Gregorio, On the set of optimal homeomorphisms for the natural pseudo-distance associated with the Lie group , Topology and its Applications, 229:187-195, 2017.