Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu’s formula

Kirill Brilliantov, Fedor Pavutnitskiy, Dmitry Pasechnyuk, German Magai
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:4542-4560, 2024.

Abstract

Computing homotopy groups of spheres has long been a fundamental objective in algebraic topology. Various theoretical and algorithmic approaches have been developed to tackle this problem. In this paper we take a step towards the goal of comprehending the group-theoretic structure of the generators of these homotopy groups by leveraging the power of machine learning. Specifically, in the simplicial group setting of Wu’s formula, we reformulate the problem of generating simplicial cycles as a problem of sampling from the intersection of algorithmic datasets related to Dyck languages. We present and evaluate language modelling approaches that employ multi-label information for input sequences, along with the necessary group-theoretic toolkit and non-neural baselines.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-brilliantov24a, title = {Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu’s formula}, author = {Brilliantov, Kirill and Pavutnitskiy, Fedor and Pasechnyuk, Dmitry and Magai, German}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {4542--4560}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/brilliantov24a/brilliantov24a.pdf}, url = {https://proceedings.mlr.press/v235/brilliantov24a.html}, abstract = {Computing homotopy groups of spheres has long been a fundamental objective in algebraic topology. Various theoretical and algorithmic approaches have been developed to tackle this problem. In this paper we take a step towards the goal of comprehending the group-theoretic structure of the generators of these homotopy groups by leveraging the power of machine learning. Specifically, in the simplicial group setting of Wu’s formula, we reformulate the problem of generating simplicial cycles as a problem of sampling from the intersection of algorithmic datasets related to Dyck languages. We present and evaluate language modelling approaches that employ multi-label information for input sequences, along with the necessary group-theoretic toolkit and non-neural baselines.} }
Endnote
%0 Conference Paper %T Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu’s formula %A Kirill Brilliantov %A Fedor Pavutnitskiy %A Dmitry Pasechnyuk %A German Magai %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-brilliantov24a %I PMLR %P 4542--4560 %U https://proceedings.mlr.press/v235/brilliantov24a.html %V 235 %X Computing homotopy groups of spheres has long been a fundamental objective in algebraic topology. Various theoretical and algorithmic approaches have been developed to tackle this problem. In this paper we take a step towards the goal of comprehending the group-theoretic structure of the generators of these homotopy groups by leveraging the power of machine learning. Specifically, in the simplicial group setting of Wu’s formula, we reformulate the problem of generating simplicial cycles as a problem of sampling from the intersection of algorithmic datasets related to Dyck languages. We present and evaluate language modelling approaches that employ multi-label information for input sequences, along with the necessary group-theoretic toolkit and non-neural baselines.
APA
Brilliantov, K., Pavutnitskiy, F., Pasechnyuk, D. & Magai, G.. (2024). Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu’s formula. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:4542-4560 Available from https://proceedings.mlr.press/v235/brilliantov24a.html.

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