Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs $(M,\backslash varphi:M\backslash to\; \backslash mathbb\{R\})\backslash$, $(N,\backslash psi:N\backslash to\; \backslash mathbb\{R\})\backslash$ is the value $\backslash inf\_h\; \backslash |\backslash varphi-\backslash psi\backslash circ\; h\backslash |\_\backslash infty\backslash$, where $h\backslash$ varies in the set of all homeomorphisms from the manifold $M\backslash$ to the manifold $N\backslash$ and $\backslash |\backslash cdot\backslash |\_\backslash infty\backslash$ is the supremum norm. If $M\backslash$ and $N\backslash$ are not homeomorphic, then the natural pseudodistance is defined to be $\backslash infty\backslash$.

It is usually assumed that $M\backslash$, $N\backslash$ are $C^1\backslash$ closed manifolds and the measuring functions $\backslash varphi,\backslash psi\backslash$ are $C^1\backslash$. Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from $M\backslash$ to $N\backslash$.

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function $\backslash varphi\backslash$ takes values in $\backslash mathbb\{R\}^m\backslash$

.Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455-464, 1999. When $M=N\backslash$, the group $H\backslash$ of all homeomorphisms of $M\backslash$ can be replaced in the definition of natural pseudodistance by a subgroup $G\backslash$ of $H\backslash$, so obtaining the concept of natural pseudodistance with respect to the group $G\backslash$.Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, 55(2):373-409, 2016.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 . Lower bounds and approximations of the natural pseudodistance with respect to the group $G\backslash$ can be obtained both by means of $G$-invariant persistent homologyPatrizio Frosini, G-invariant persistent homology, Mathematical Methods in the Applied Sciences, 38(6):1190-1199, 2015. and by combining classical persistent homology with the use of G-equivariant non-expansive operators.Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, 55(2):373-409, 2016.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 .