Natural pseudodistance

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 ${\displaystyle \mathbb {R} ^{m}\ }$ .[1] 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 ${\displaystyle G\ }$.[2][3] 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[4] and by combining classical persistent homology with the use of G-equivariant non-expansive operators.[2][3]

Main properties

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 ${\displaystyle k\ }$. If ${\displaystyle M\ }$ and ${\displaystyle N\ }$ are surfaces, the number ${\displaystyle k\ }$ can be assumed to be ${\displaystyle 1\ }$, ${\displaystyle 2\ }$ or ${\displaystyle 3\ }$.[6] If ${\displaystyle M\ }$ and ${\displaystyle N\ }$ are curves, the number ${\displaystyle k\ }$ can be assumed to be ${\displaystyle 1\ }$ or ${\displaystyle 2\ }$.[7] 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 ${\displaystyle 1\ }$.[5] The research concerning optimal homeomorphisms is still at its very beginning .[8][9]

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.