TRIPTURES-TRITTURE

NATURAL PSEUDO-DISTANCE WITH RESPECT TO A GROUP G

The natural pseudo-distance d_G (φ,ψ) between two continuous functions φ,ψ:X → ℝ^k is the supremum of the max-norm of the function φ-ψ⚬ g, when g varies in the group G, provided that G is a group of self-homeomorphisms of X. The rationale of the natural pseudo-distance d_G is that it is sometimes of use to compare data with respect to the invariance expressed by a group G: for example, if two images represented by two functions φ,ψ:ℝ² → ℝ³ are obtained by a translation t of the real plane (i.e., φ=ψ⚬ t), the natural pseudo-metric d_G makes it possible to say that the difference between these two images vanishes. This is not only useful for applications, but also from a theoretical point of view (e.g., if we are interested in the comparison of Riemannian structures represented by functions defined on a suitable fiber bundle, when an invariance group is involved).

If you are interested in the natural pseudo-distance d_G, please have a look at the paper [2].

For a quick description of the concept and for open problems, please also have a look at these slides:

International Workshop and Conference on Topology & Applications, Rajagiri School of Engineering & Technology, Kochi, Kerala, INDIA, December 5-11, 2018,
The natural pseudo-distance in topological data analysis.

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REFERENCES

[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, vol. 1, n. 9, pages 423–433 (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.

[2] Patrizio Frosini, Grzegorz Jabłoński, Combining persistent homology and invariance groups for shape comparison, Discrete & Computational Geometry, vol. 55 (2016), n. 2, pages 373-409. DOI: 10.1007/s00454-016-9761-y. Preprint available at http://arxiv.org/pdf/1312.7219v4.pdf. Paper available at http://link.springer.com/article/10.1007/s00454-016-9761-y.

[3] Michele d'Amico, Patrizio Frosini, Claudia Landi, Natural pseudo-distance and optimal matching between reduced size functions, Acta Applicandae Mathematicae, vol. 109 (2010), n. 2, 527-554. Preprint (Some results in this paper, including the proof of stability of persistence diagrams in degree 0, had already appeared in this technical report:

Michele d'Amico, Patrizio Frosini, Claudia Landi, Optimal matching between reduced size functions, Technical report no. 35 (2003), DISMI, University of Modena and Reggio Emilia, Italy.)

[4] Pietro Donatini, Patrizio Frosini, Natural pseudo-distances between closed curves, Forum Mathematicum, vol. 21 (2009), Issue 6, 981–999. Preprint

[5] Pietro Donatini, Patrizio Frosini, Natural pseudo-distances between closed surfaces, Journal of the European Mathematical Society, vol. 9 (2007), n. 2, 331–353. Preprint

[6] Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, vol. 16 (2004), n. 5, 695-715. Preprint

[7] Pietro Donatini, Patrizio Frosini, Lower bounds for natural pseudodistances via size functions, Archives of Inequalities and Applications, vol. 2 (2004), n. 1, 1-12. Preprint

[8] Patrizio Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bulletin of the Australian Mathematical Society, 42, 3 (1990), 407-416.