Interpreting deep neural networks trained on elementary $p$ groups reveals algorithmic structure

Gavin McCracken, Arthur Ayestas Hilgert, Sihui Wei, Gabriela Moisescu-Pareja, Zhaoyue Wang, Jonathan Love
Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025), PMLR 321:389-402, 2026.

Abstract

We interpret deep neural networks (DNNs) trained on elementary $p$ group multiplication, examining how our results reveal some of the nature within major deep learning hypotheses. Assisted by tools from computational algebra and geometry, we perform analyses at multiple levels of abstraction, finding we can fully characterize and describe: 1) the global algorithm DNNs learn on this task—the multidimensional Chinese remainder theorem; 2) the neural representations, which are 2-torus $\mathbb{T}^2$ embedded in $\mathbb{R}^4$ encoding coset structure; 3) the individual neuron activation patterns, which activate solely on coset structures of the group. Furthermore, we find neurons learn the Lee metric to organize their activation strengths. Overall, our work serves as an exposition toward understanding how DNNs learn group multiplications.

Cite this Paper

BibTeX
@InProceedings{pmlr-v321-mccracken26a, title = {Interpreting deep neural networks trained on elementary $p$ groups reveals algorithmic structure}, author = {McCracken, Gavin and Ayestas Hilgert, Arthur and Wei, Sihui and Moisescu-Pareja, Gabriela and Wang, Zhaoyue and Love, Jonathan}, booktitle = {Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025)}, pages = {389--402}, year = {2026}, editor = {Bernardez Gil, Guillermo and Black, Mitchell and Cloninger, Alexander and Doster, Timothy and Emerson, Tegan and Garcı́a-Rodondo, Ińes and Holtz, Chester and Kotak, Mit and Kvinge, Henry and Mishne, Gal and Papillon, Mathilde and Pouplin, Alison and Rainey, Katie and Rieck, Bastian and Telyatnikov, Lev and Yeats, Eric and Wang, Qingsong and Wang, Yusu and Wayland, Jeremy}, volume = {321}, series = {Proceedings of Machine Learning Research}, month = {01--02 Dec}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v321/main/assets/mccracken26a/mccracken26a.pdf}, url = {https://proceedings.mlr.press/v321/mccracken26a.html}, abstract = {We interpret deep neural networks (DNNs) trained on elementary $p$ group multiplication, examining how our results reveal some of the nature within major deep learning hypotheses. Assisted by tools from computational algebra and geometry, we perform analyses at multiple levels of abstraction, finding we can fully characterize and describe: 1) the global algorithm DNNs learn on this task—the multidimensional Chinese remainder theorem; 2) the neural representations, which are 2-torus $\mathbb{T}^2$ embedded in $\mathbb{R}^4$ encoding coset structure; 3) the individual neuron activation patterns, which activate solely on coset structures of the group. Furthermore, we find neurons learn the Lee metric to organize their activation strengths. Overall, our work serves as an exposition toward understanding how DNNs learn group multiplications.} }
Endnote
%0 Conference Paper %T Interpreting deep neural networks trained on elementary $p$ groups reveals algorithmic structure %A Gavin McCracken %A Arthur Ayestas Hilgert %A Sihui Wei %A Gabriela Moisescu-Pareja %A Zhaoyue Wang %A Jonathan Love %B Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025) %C Proceedings of Machine Learning Research %D 2026 %E Guillermo Bernardez Gil %E Mitchell Black %E Alexander Cloninger %E Timothy Doster %E Tegan Emerson %E Ińes Garcı́a-Rodondo %E Chester Holtz %E Mit Kotak %E Henry Kvinge %E Gal Mishne %E Mathilde Papillon %E Alison Pouplin %E Katie Rainey %E Bastian Rieck %E Lev Telyatnikov %E Eric Yeats %E Qingsong Wang %E Yusu Wang %E Jeremy Wayland %F pmlr-v321-mccracken26a %I PMLR %P 389--402 %U https://proceedings.mlr.press/v321/mccracken26a.html %V 321 %X We interpret deep neural networks (DNNs) trained on elementary $p$ group multiplication, examining how our results reveal some of the nature within major deep learning hypotheses. Assisted by tools from computational algebra and geometry, we perform analyses at multiple levels of abstraction, finding we can fully characterize and describe: 1) the global algorithm DNNs learn on this task—the multidimensional Chinese remainder theorem; 2) the neural representations, which are 2-torus $\mathbb{T}^2$ embedded in $\mathbb{R}^4$ encoding coset structure; 3) the individual neuron activation patterns, which activate solely on coset structures of the group. Furthermore, we find neurons learn the Lee metric to organize their activation strengths. Overall, our work serves as an exposition toward understanding how DNNs learn group multiplications.
APA
McCracken, G., Ayestas Hilgert, A., Wei, S., Moisescu-Pareja, G., Wang, Z. & Love, J.. (2026). Interpreting deep neural networks trained on elementary $p$ groups reveals algorithmic structure. Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025), in Proceedings of Machine Learning Research 321:389-402 Available from https://proceedings.mlr.press/v321/mccracken26a.html.

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