{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:31:09Z","timestamp":1753893069811,"version":"3.41.2"},"reference-count":24,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T00:00:00Z","timestamp":1703116800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Artif. Intell."],"abstract":"<jats:p>In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. If the set of transformations acting is a group, we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subjected to the action of certain self-maps and the space of P-GENEOs between these spaces. We define pseudo-metrics on them and show some properties of the resulting spaces. In particular, we show how such spaces have convenient approximation and convexity properties.<\/jats:p>","DOI":"10.3389\/frai.2023.1272619","type":"journal-article","created":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T04:47:34Z","timestamp":1703134054000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["A topological model for partial equivariance in deep learning and data analysis"],"prefix":"10.3389","volume":"6","author":[{"given":"Lucia","family":"Ferrari","sequence":"first","affiliation":[]},{"given":"Patrizio","family":"Frosini","sequence":"additional","affiliation":[]},{"given":"Nicola","family":"Quercioli","sequence":"additional","affiliation":[]},{"given":"Francesca","family":"Tombari","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2023,12,21]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1038\/s42256-019-0087-3","article-title":"Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning","volume":"1","author":"Bergomi","year":"2019","journal-title":"Nat. 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