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Pullback Bundles and the Geometry of Learning
Explainable Artificial Intelligence (XAI) and acceptable artificial intelligence are active topics of research in machine learning. For critical applications, being able to prove or at least to ensure with a high probability the correctness of algorithms is of utmost importance. In practice, however...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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MDPI
2023
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10606266/ https://www.ncbi.nlm.nih.gov/pubmed/37895571 http://dx.doi.org/10.3390/e25101450 |
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author | Puechmorel, Stéphane |
author_facet | Puechmorel, Stéphane |
author_sort | Puechmorel, Stéphane |
collection | PubMed |
description | Explainable Artificial Intelligence (XAI) and acceptable artificial intelligence are active topics of research in machine learning. For critical applications, being able to prove or at least to ensure with a high probability the correctness of algorithms is of utmost importance. In practice, however, few theoretical tools are known that can be used for this purpose. Using the Fisher Information Metric (FIM) on the output space yields interesting indicators in both the input and parameter spaces, but the underlying geometry is not yet fully understood. In this work, an approach based on the pullback bundle, a well-known trick for describing bundle morphisms, is introduced and applied to the encoder–decoder block. With constant rank hypothesis on the derivative of the network with respect to its inputs, a description of its behavior is obtained. Further generalization is gained through the introduction of the pullback generalized bundle that takes into account the sensitivity with respect to weights. |
format | Online Article Text |
id | pubmed-10606266 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2023 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
spelling | pubmed-106062662023-10-28 Pullback Bundles and the Geometry of Learning Puechmorel, Stéphane Entropy (Basel) Article Explainable Artificial Intelligence (XAI) and acceptable artificial intelligence are active topics of research in machine learning. For critical applications, being able to prove or at least to ensure with a high probability the correctness of algorithms is of utmost importance. In practice, however, few theoretical tools are known that can be used for this purpose. Using the Fisher Information Metric (FIM) on the output space yields interesting indicators in both the input and parameter spaces, but the underlying geometry is not yet fully understood. In this work, an approach based on the pullback bundle, a well-known trick for describing bundle morphisms, is introduced and applied to the encoder–decoder block. With constant rank hypothesis on the derivative of the network with respect to its inputs, a description of its behavior is obtained. Further generalization is gained through the introduction of the pullback generalized bundle that takes into account the sensitivity with respect to weights. MDPI 2023-10-15 /pmc/articles/PMC10606266/ /pubmed/37895571 http://dx.doi.org/10.3390/e25101450 Text en © 2023 by the author. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Puechmorel, Stéphane Pullback Bundles and the Geometry of Learning |
title | Pullback Bundles and the Geometry of Learning |
title_full | Pullback Bundles and the Geometry of Learning |
title_fullStr | Pullback Bundles and the Geometry of Learning |
title_full_unstemmed | Pullback Bundles and the Geometry of Learning |
title_short | Pullback Bundles and the Geometry of Learning |
title_sort | pullback bundles and the geometry of learning |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10606266/ https://www.ncbi.nlm.nih.gov/pubmed/37895571 http://dx.doi.org/10.3390/e25101450 |
work_keys_str_mv | AT puechmorelstephane pullbackbundlesandthegeometryoflearning |