Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently non-Euclidean. This data can exhibit intr...
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| Main Authors: | , , , , , , , , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2025-01-01
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| Series: | Machine Learning: Science and Technology |
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| Online Access: | https://doi.org/10.1088/2632-2153/adf375 |
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| author | Mathilde Papillon Sophia Sanborn Johan Mathe Louisa Cornelis Abby Bertics Domas Buracas Hansen J Lillemark Christian Shewmake Fatih Dinc Xavier Pennec Nina Miolane |
| author_facet | Mathilde Papillon Sophia Sanborn Johan Mathe Louisa Cornelis Abby Bertics Domas Buracas Hansen J Lillemark Christian Shewmake Fatih Dinc Xavier Pennec Nina Miolane |
| author_sort | Mathilde Papillon |
| collection | DOAJ |
| description | The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently non-Euclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field. |
| format | Article |
| id | doaj-art-dbc648794d4c4bf08efe0843424d5f3d |
| institution | Matheson Library |
| issn | 2632-2153 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | Machine Learning: Science and Technology |
| spelling | doaj-art-dbc648794d4c4bf08efe0843424d5f3d2025-08-01T12:51:28ZengIOP PublishingMachine Learning: Science and Technology2632-21532025-01-016303100210.1088/2632-2153/adf375Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structuresMathilde Papillon0https://orcid.org/0000-0003-1674-4218Sophia Sanborn1https://orcid.org/0000-0002-1957-7067Johan Mathe2https://orcid.org/0009-0000-8096-574XLouisa Cornelis3https://orcid.org/0009-0000-7156-9884Abby Bertics4https://orcid.org/0009-0000-5081-4983Domas Buracas5Hansen J Lillemark6Christian Shewmake7https://orcid.org/0000-0003-4363-5615Fatih Dinc8https://orcid.org/0000-0003-0921-0162Xavier Pennec9https://orcid.org/0000-0002-6617-7664Nina Miolane10https://orcid.org/0000-0002-1200-9024UC Santa Barbara , Santa Barbara, United States of America; Equal contributionStanford University , Palo Alto, United States of America; Equal contributionAtmo, Inc., San Francisco , United States of America; Equal contributionUC Santa Barbara , Santa Barbara, United States of America; Equal contributionUC Santa Barbara , Santa Barbara, United States of AmericaNew Theory AI , San Francisco, United States of AmericaNew Theory AI , San Francisco, United States of America; UC Berkeley , Berkeley, United States of AmericaNew Theory AI , San Francisco, United States of AmericaUC Santa Barbara , Santa Barbara, United States of AmericaUniversité Côte d’Azur & Inria , Nice, FranceUC Santa Barbara , Santa Barbara, United States of America; Stanford University , Palo Alto, United States of America; Atmo, Inc., San Francisco , United States of America; New Theory AI , San Francisco, United States of AmericaThe enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently non-Euclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.https://doi.org/10.1088/2632-2153/adf375geometric deep learninggeometrytopologyalgebramachine learning |
| spellingShingle | Mathilde Papillon Sophia Sanborn Johan Mathe Louisa Cornelis Abby Bertics Domas Buracas Hansen J Lillemark Christian Shewmake Fatih Dinc Xavier Pennec Nina Miolane Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures Machine Learning: Science and Technology geometric deep learning geometry topology algebra machine learning |
| title | Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures |
| title_full | Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures |
| title_fullStr | Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures |
| title_full_unstemmed | Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures |
| title_short | Beyond Euclid: an illustrated guide to modern machine learning with geometric, topological, and algebraic structures |
| title_sort | beyond euclid an illustrated guide to modern machine learning with geometric topological and algebraic structures |
| topic | geometric deep learning geometry topology algebra machine learning |
| url | https://doi.org/10.1088/2632-2153/adf375 |
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