# Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

We provide a general mathematical framework for group and set equivariance in machine learning. We define group equivariant
non-expansive operators (GENEOs) as maps between function spaces associated with groups of transformations. We study the
topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general
strategies to initialize and compose operators. We define suitable pseudo-metrics for the function spaces, the equivariance
groups and the set of non-expansive operators. We prove that, under suitable assumptions, the space of GENEOs is compact and
convex. These results provide fundamental guarantees in a machine learning perspective. By considering isometry-equivariant
non-expansive operators, we describe a simple strategy to select and sample operators. Thereafter, we show how selected and
sampled operators can be used both to perform classical metric learning and to inject knowledge in artificial neural networks.

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## BibTex references

@Article\{BFGQ19, author = "Bergomi, Mattia G. and Frosini, Patrizio and Giorgi, Daniela and Quercioli, Nicola", title = "Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning", journal = "Nature Machine Intelligence", volume = "1", pages = "423-433", month = "sep", year = "2019", url = "http://vcg.isti.cnr.it/Publications/2019/BFGQ19" }