Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915315005259776 |
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| author | Marchetti, Giovanni Luca Shahverdi, Vahid Mereta, Stefano Trager, Matthew Kohn, Kathlén |
| author_facet | Marchetti, Giovanni Luca Shahverdi, Vahid Mereta, Stefano Trager, Matthew Kohn, Kathlén |
| contents | In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and discuss ideas beyond the algebraic domain. This work lays the foundations of a research direction bridging algebraic geometry and deep learning, that we refer to as neuroalgebraic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry Marchetti, Giovanni Luca Shahverdi, Vahid Mereta, Stefano Trager, Matthew Kohn, Kathlén Machine Learning Algebraic Geometry In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and discuss ideas beyond the algebraic domain. This work lays the foundations of a research direction bridging algebraic geometry and deep learning, that we refer to as neuroalgebraic geometry. |
| title | Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry |
| topic | Machine Learning Algebraic Geometry |
| url | https://arxiv.org/abs/2501.18915 |