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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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Fig. 1: Isometry equivariant non-expansive operators on MNIST.
Fig. 2: Persistent homology background.
Fig. 3: Experimental pipeline.
Fig. 4: IENEO-based metric learning on MNIST.
Fig. 5: IENEO-based metric learning on fashion-MNIST and CIFAR-10.
Fig. 6: Metric learning on transformed MNIST validation samples.
Fig. 7: Convolutional neural network architecture used in the knowledge injection experiment.

Code availability

The experiments and applications can be reproduced by installing the open-source Python package, available at

Data availability

No datasets were generated during the current study. The datasets analysed are available at





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The authors thank H. Young for proofreading the manuscript. The research carried out by M.G.B. was supported by the European Research Council (Advanced Investigator Grant 671251 to Z.F. Mainen), the Champalimaud Foundation (Z.F. Mainen) and a GPU NVIDIA grant. The research carried out by P.F. and N.Q. was partially supported by GNSAGA-INdAM (Italy).

Author information




P.F., M.G.B., D.G. and N.Q. devised the project and algorithms. P.F. and N.Q. formalized the mathematical model with help from M.G.B. and D.G. M.G.B. wrote the code with help from D.G. M.G.B. and D.G. analysed the data. P.F. supervised the project. M.G.B., P.F., D.G. and N.Q. contributed equally to the writing of the main text and methods.

Corresponding author

Correspondence to Patrizio Frosini.

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The authors declare no competing interests.

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Bergomi, M.G., Frosini, P., Giorgi, D. et al. Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning. Nat Mach Intell 1, 423–433 (2019).

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