Natural pseudodistance
In size theory, the natural pseudodistance between two size pairs [math]\displaystyle{ (M,\varphi:M\to \mathbb{R})\ }[/math], [math]\displaystyle{ (N,\psi:N\to \mathbb{R})\ }[/math] is the value [math]\displaystyle{ \inf_h \|\varphi-\psi\circ h\|_\infty\ }[/math], where [math]\displaystyle{ h\ }[/math] varies in the set of all homeomorphisms from the manifold [math]\displaystyle{ M\ }[/math] to the manifold [math]\displaystyle{ N\ }[/math] and [math]\displaystyle{ \|\cdot\|_\infty\ }[/math] is the supremum norm. If [math]\displaystyle{ M\ }[/math] and [math]\displaystyle{ N\ }[/math] are not homeomorphic, then the natural pseudodistance is defined to be [math]\displaystyle{ \infty\ }[/math]. It is usually assumed that [math]\displaystyle{ M\ }[/math], [math]\displaystyle{ N\ }[/math] are [math]\displaystyle{ C^1\ }[/math] closed manifolds and the measuring functions [math]\displaystyle{ \varphi,\psi\ }[/math] are [math]\displaystyle{ C^1\ }[/math]. Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from [math]\displaystyle{ M\ }[/math] to [math]\displaystyle{ N\ }[/math].
The concept of natural pseudodistance can be easily extended to size pairs where the measuring function [math]\displaystyle{ \varphi\ }[/math] takes values in [math]\displaystyle{ \mathbb{R}^m\ }[/math] .[1] When [math]\displaystyle{ M=N\ }[/math], the group [math]\displaystyle{ H\ }[/math] of all homeomorphisms of [math]\displaystyle{ M\ }[/math] can be replaced in the definition of natural pseudodistance by a subgroup [math]\displaystyle{ G\ }[/math] of [math]\displaystyle{ H\ }[/math], so obtaining the concept of natural pseudodistance with respect to the group [math]\displaystyle{ G\ }[/math].[2][3] Lower bounds and approximations of the natural pseudodistance with respect to the group [math]\displaystyle{ G\ }[/math] can be obtained both by means of [math]\displaystyle{ G }[/math]-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 [math]\displaystyle{ k\ }[/math]. If [math]\displaystyle{ M\ }[/math] and [math]\displaystyle{ N\ }[/math] are surfaces, the number [math]\displaystyle{ k\ }[/math] can be assumed to be [math]\displaystyle{ 1\ }[/math], [math]\displaystyle{ 2\ }[/math] or [math]\displaystyle{ 3\ }[/math].[6] If [math]\displaystyle{ M\ }[/math] and [math]\displaystyle{ N\ }[/math] are curves, the number [math]\displaystyle{ k\ }[/math] can be assumed to be [math]\displaystyle{ 1\ }[/math] or [math]\displaystyle{ 2\ }[/math].[7] If an optimal homeomorphism [math]\displaystyle{ \bar h\ }[/math] exists (i.e., [math]\displaystyle{ \|\varphi-\psi\circ \bar h\|_\infty=\inf_h \|\varphi-\psi\circ h\|_\infty\ }[/math]), then [math]\displaystyle{ k\ }[/math] can be assumed to be [math]\displaystyle{ 1\ }[/math].[5] The research concerning optimal homeomorphisms is still at its very beginning .[8][9]
See also
References
- ↑ Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455-464, 1999.
- ↑ 2.0 2.1 Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, 55(2):373-409, 2016.
- ↑ 3.0 3.1 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 .
- ↑ Patrizio Frosini, G-invariant persistent homology, Mathematical Methods in the Applied Sciences, 38(6):1190-1199, 2015.
- ↑ 5.0 5.1 Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.
- ↑ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.
- ↑ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.
- ↑ Andrea Cerri, Barbara Di Fabio, On certain optimal diffeomorphisms between closed curves, Forum Mathematicum, 26(6):1611-1628, 2014.
- ↑ Alessandro De Gregorio, On the set of optimal homeomorphisms for the natural pseudo-distance associated with the Lie group [math]\displaystyle{ S^1\ }[/math], Topology and its Applications, 229:187-195, 2017.
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