On the Role of Group Equivariant Non-Expansive Operators as a Bridge between TDA and Machine Learning

Abstract: Group Equivariant Non-Expansive Operators (GENEOs) were introduced ten years ago as a tool to reduce and modulate the invariance of persistence diagrams (originally valid for every reparameterization of the signal domain) [1]. The computation of persistence diagrams itself can be seen as an example of a GENEO. Subsequently, these operators have been independently studied and employed in various applications in data analysis and machine learning [2-7]. In this talk, we will illustrate the definitions and basic properties of the main concepts used in GENEO theory, while also highlighting their promising applications in TDA and Explainable Artificial Intelligence.

[1] 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.

[2] 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.

[3] Giovanni Bocchi, Stefano Botteghi, Martina Brasini, Patrizio Frosini, 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.

[4] Giovanni Bocchi, Patrizio Frosini, Massimo Ferri, A novel approach to graph distinction through GENEOs and permutants, Scientific Reports, 15 (2025), 6259. DOI: 10.1038/s41598-025-90152-7.

[5] Giovanni Bocchi, Patrizio Frosini, Alessandra Micheletti, Alessandro Pedretti, Gianluca Palermo, Davide Gadioli, Carmen Gratteri, Filippo Lunghini, Akash Deep Biswas, Pieter F.W. Stouten, Andrea R. Beccari, Anna Fava, Carmine Talarico, GENEOnet: A breakthrough in protein binding pocket detection using group equivariant non-expansive operators, Scientific Reports, 15 (2025), 34597. DOI:10.1038/s41598-025-18132-5.

[6] Raúl Felipe, GENEOs with respect to the projective Hilbert metric, The Journal of Geometric Analysis, vol. 35 (9) (2025), 264. DOI: 10.1007/s12220-025-02102-4.

[7] Diogo Lavado, Alessandra Micheletti, Giovanni Bocchi, Patrizio Frosini, Cláudia Soares, SCENE-Net: Geometric induction for interpretable and low-resource 3D pole detection with Group-Equivariant Non-Expansive Operators, Computer Vision and Image Understanding, vol. 262 (2025), 104531. DOI: 10.1016/j.cviu.2025.104531.

geometric topology

Audience: researchers in the topic

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GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology

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