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A topological model for partial equivariance in deep learning and data analysis.

Lucia Ferrari1, Patrizio Frosini1, Nicola Quercioli2

  • 1Department of Mathematics, University of Bologna, Bologna, Italy.

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Summary
This summary is machine-generated.

We introduce a topological model for partial equivariance in neural networks using P-GENEO operators. These operators ensure data transformations respect certain symmetries, offering approximation and convexity properties for enhanced network performance.

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P-GENEOcompactnessconvexitypartial-equivariant neural networkpseudo-metric space

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Area of Science:

  • * Mathematics
  • * Computer Science
  • * Machine Learning

Background:

  • * Neural networks often require data transformations that respect underlying symmetries for improved performance and generalization.
  • * Encoding partial equivariance, where transformations are not necessarily groups, presents a significant challenge in network design.

Purpose of the Study:

  • * To propose a novel topological model for encoding partial equivariance in neural networks.
  • * To introduce and analyze a new class of operators, P-GENEOs, for data transformation.
  • * To investigate the properties of spaces of measurements and P-GENEOs.

Main Methods:

  • * Development of a topological framework to model partial equivariance.
  • * Introduction of P-GENEOs (Partial-set General Equivariant Network Operators) as data transformation operators.
  • * Mathematical analysis of spaces of measurements and P-GENEOs, including definition of pseudo-metrics.

Main Results:

  • * P-GENEOs are defined as non-expansive operators respecting transformations from specific sets.
  • * GENEOs (General Equivariant Network Operators) are a special case when transformations form a group.
  • * The study demonstrates that the resulting spaces of measurements and P-GENEOs possess convenient approximation and convexity properties.

Conclusions:

  • * The proposed topological model effectively encodes partial equivariance in neural networks.
  • * P-GENEOs provide a flexible tool for handling data transformations with partial symmetries.
  • * The identified approximation and convexity properties are beneficial for the theoretical understanding and practical application of these networks.