TRIPTURES-TRITTURE

GROUP EQUIVARIANT NON-EXPANSIVE OPERATORS (GENEOS)

In machine learning agents acting on data can be often seen as group equivariant non-expansive operators (GENEOs). Therefore, approximating an agent A means to approximate A in the metric space of GENEOs. Persistent homology enters this framework as a an efficient mathematical tool that makes available both a pseudo-distance between data and a proxy pseudo-distance between GENEOs. This is a key point since, without persistent homology, the metric comparison of GENEOs would be computationally too expensive. A detailed description of this approach can be found in the paper [1].

If you are interested in the use of group equivariant non-expansive operators and persistent homology for machine learning, please have a look at these slides:

10th Conference on Geometric and Topological Methods in Computer Science (GETCO), Oaxaca, Mexico, September 10-14, 2018,
The theory of group equivariant non-expansive operators in topological data analysis.

I have to confess that working with operators is not trivial. If you were born in the '60s and like songs, this link could be of use ;-).

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REFERENCES

[1] Lucia Ferrari, Patrizio Frosini, Nicola Quercioli, Francesca Tombari,
A topological model for partial equivariance in deep learning and data analysis,
Frontiers in Artificial Intelligence, 6 (2023). DOI: 10.3389/frai.2023.1272619. OPEN ACCESS URL: https://www.frontiersin.org/articles/10.3389/frai.2023.1272619 .

[2] Faraz Ahmad, Massimo Ferri, Patrizio Frosini,
Generalized Permutants and Graph GENEOs,
Machine Learning and Knowledge Extraction, 5(4) (2023), 1905-1920. DOI: 10.3390/make5040092. OPEN ACCESS URL: https://www.mdpi.com/2504-4990/5/4/92 .

[3] Alessandra Micheletti,
A new paradigm for artificial intelligence based on group equivariant non-expansive operators,
European Mathematical Society Magazine, vol. 128 (2023), 4–12. DOI 10.4171/MAG/133. OPEN ACCESS URL: https://euromathsoc.org/magazine/articles/133 .

[4] Patrizio Frosini, Ivan Gridelli, Andrea Pascucci,
A probabilistic result on impulsive noise reduction in topological data analysis through group equivariant non-expansive operators,
Special issue "Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems."
Entropy, vol. 25 (2023), Issue 8, 1150. OPEN ACCESS URL: https://www.mdpi.com/1099-4300/25/8/1150 .

[5] Giovanni Bocchi, Stefano Botteghi, Martina Brasini, Patrizio Frosini and Nicola Quercioli,
On the finite representation of linear group equivariant operators via permutant measures,
Annals of Mathematics and Artificial Intelligence, vol. 91 (2023), n. 4, 465–487. DOI: 10.1007/s10472-022-09830-1. OPEN ACCESS URL: https://link.springer.com/article/10.1007/s10472-022-09830-1 . Full-text access to a view-only version of the paper: https://rdcu.be/c5Obw .

[6] Francesco Conti, Patrizio Frosini, Nicola Quercioli,
On the Construction of Group Equivariant Non-Expansive Operators via Permutants and Symmetric Functions,
Frontiers in Artificial Intelligence, vol. 5 (2022), 1-11. DOI: 10.3389/frai.2022.786091. OPEN ACCESS URL: https://www.frontiersin.org/article/10.3389/frai.2022.786091.

[7] 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.

[8] Patrizio Frosini, Grzegorz Jabłoński,
Combining persistent homology and invariance groups for shape comparison,
Discrete & Computational Geometry, vol. 55 (2016), n. 2, 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.

[9] Patrizio Frosini, Nicola Quercioli,
Some remarks on the algebraic properties of group invariant operators in persistent homology,
Lecture Notes in Computer Science, Proceedings of the International Cross-Domain Conference, CD-MAKE 2017, Reggio, Italy, August 29–September 1, 2017, MAKE Topology, Springer, Cham, Holzinger A., Kieseberg P., Tjoa A M., Weippl E. (Eds.), LNCS 10410, 14-24, 2017. DOI: 10.1007/978-3-319-66808-6_2. Preprint available here.

[10] Francesco Camporesi, Patrizio Frosini, Nicola Quercioli,
On a new method to build group equivariant operators by means of permutants,
Lecture Notes in Computer Science, Proceedings of the International Cross-Domain Conference, CD-MAKE 2018, Hamburg, Germany, August 27-30, 2018, MAKE Topology, Springer, Cham, A. Holzinger et al. (Eds.), LNCS 11015, 265–272, 2018. DOI: 10.1007/978-3-319-99740-7_18. Preprint available here.

[11] Patrizio Frosini,
Towards an observer-oriented theory of shape comparison,
Proceedings of the 8th Eurographics Workshop on 3D Object Retrieval, Lisbon, Portugal, May 7-8, 2016, A. Ferreira, A. Giachetti, and D. Giorgi (Editors), 5-8. DOI: 10.2312/3dor.20161080. Preprint available at http://arxiv.org/pdf/arXiv:1603.02008v1.pdf.