Natural pseudodistance explained

In size theory, the natural pseudodistance between two size pairs

(M,\varphi:M\to R ) , (N,\psi:N\to R ) is the value inf _{h} \|\varphi - \psi\circ h\| _{infty} , where h varies in the set of all homeomorphisms from the manifold M to the manifold N and \| ⋅ \| _{infty} is the supremum norm. If M and N are not homeomorphic, then the natural pseudodistance is defined to be inf ty .It is usually assumed that M , N are C ^{1} closed manifolds and the measuring functions \varphi,\psi are C ^{1} . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from M to N .

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function

\varphi takes values in R ^{m} .^[1] When M = N , the group H of all homeomorphisms of M can be replaced in the definition of natural pseudodistance by a subgroup G of H , so obtaining the concept of natural pseudodistance with respect to the group G .^{[2] [3]} Lower bounds and approximations of the natural pseudodistance with respect to the group G can be obtained both by means of G -invariant persistent homology^[4] and by combining classical persistent homology with the use of G-equivariant non-expansive operators.^{[5] [6]}

Main properties

It can be proved ^[7] 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

k .If M and N are surfaces, the number k can be assumed to be 1 , 2 or 3 .^[8] If M and N are curves, the number k can be assumed to be 1 or 2 .^[9] If an optimal homeomorphism \bar h exists (i.e., \|\varphi - \psi\circ \bar h\| _{infty=inf} _{h} \|\varphi - \psi\circ h\| _{infty} ), then k can be assumed to be 1 .^[7] The research concerning optimal homeomorphisms is still at its very beginning.^{[10] [11]}

See also

• Fréchet distance
• Size function
• Size functor
• Size homotopy group

Notes and References

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. Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, 55(2):373-409, 2016.
3. 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. Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, 55(2):373-409, 2016.
6. 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 .
7. Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.
8. Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces,Journal of the European Mathematical Society, 9(2):231–253, 2007.
9. Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.
10. Andrea Cerri, Barbara Di Fabio, On certain optimal diffeomorphisms between closed curves, Forum Mathematicum, 26(6):1611-1628, 2014.
11. Alessandro De Gregorio, On the set of optimal homeomorphisms for the natural pseudo-distance associated with the Lie group S ^{1} ,Topology and its Applications, 229:187-195, 2017.