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

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function ${\displaystyle \varphi \ }$ takes values in ${\displaystyle \mathbb {R} ^{m}\ }$ . When ${\displaystyle M=N\ }$, the group ${\displaystyle H\ }$ of all homeomorphisms of ${\displaystyle M\ }$ can be replaced in the definition of natural pseudodistance by a subgroup ${\displaystyle G\ }$ of ${\displaystyle H\ }$, so obtaining the concept of natural pseudodistance with respect to the group ${\displaystyle G\ }$. Lower bounds and approximations of the natural pseudodistance with respect to the group ${\displaystyle G\ }$ can be obtained both by means of ${\displaystyle G}$-invariant persistent homology and by combining classical persistent homology with the use of G-equivariant non-expansive operators.

Main properties

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

See also

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

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. ^ 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 ${\displaystyle S^{1}\ }$, Topology and its Applications, 229:187-195, 2017.

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