Position: Algebra Unveils Deep Learning - An Invitation to Neuroalgebraic Geometry

Giovanni Luca Marchetti, Vahid Shahverdi, Stefano Mereta, Matthew Trager, Kathlén Kohn
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:81759-81773, 2025.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-marchetti25a, title = {Position: Algebra Unveils Deep Learning - An Invitation to Neuroalgebraic Geometry}, author = {Marchetti, Giovanni Luca and Shahverdi, Vahid and Mereta, Stefano and Trager, Matthew and Kohn, Kathl\'{e}n}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {81759--81773}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/marchetti25a/marchetti25a.pdf}, url = {https://proceedings.mlr.press/v267/marchetti25a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Position: Algebra Unveils Deep Learning - An Invitation to Neuroalgebraic Geometry %A Giovanni Luca Marchetti %A Vahid Shahverdi %A Stefano Mereta %A Matthew Trager %A Kathlén Kohn %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-marchetti25a %I PMLR %P 81759--81773 %U https://proceedings.mlr.press/v267/marchetti25a.html %V 267 %X 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.
APA
Marchetti, G.L., Shahverdi, V., Mereta, S., Trager, M. & Kohn, K.. (2025). Position: Algebra Unveils Deep Learning - An Invitation to Neuroalgebraic Geometry. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:81759-81773 Available from https://proceedings.mlr.press/v267/marchetti25a.html.

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