Tracking Representation Dynamics in Large Language Models with Persistent Homology

Naman Malhotra, Jay Ambadkar, Abhinav Gupta, Kushal Kasivel, Abbas Schwarz, Kamillo Ferry, Anthea Monod
Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026), PMLR 334(2):211-245, 2026.

Abstract

Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.

Cite this Paper

BibTeX
@InProceedings{pmlr-v334-malhotra26a, title = {Tracking Representation Dynamics in Large Language Models with Persistent Homology}, author = {Malhotra, Naman and Ambadkar, Jay and Gupta, Abhinav and Kasivel, Kushal and Schwarz, Abbas and Ferry, Kamillo and Monod, Anthea}, booktitle = {Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026)}, pages = {211--245}, year = {2026}, editor = {Berman, Eddie and Bernárdez, Guillermo and Chen, Samantha and Cloninger, Alex and Doster, Timothy and Emerson, Tegan and Grigsby, J. Elisenda and Kvinge, Henry and Lawrence, Hannah and Marrinan, Tim and Myers, Audun and Papillon, Mathilde and Tahmasebi, Behrooz and Telyatnikov, Lev and Walters, Robin and Weber, Melanie and Xie, YuQing and Yeats, Eric}, volume = {334}, number = {2}, series = {Proceedings of Machine Learning Research}, month = {18--20 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v334/main/assets/malhotra26a/malhotra26a.pdf}, url = {https://proceedings.mlr.press/v334/malhotra26a.html}, abstract = {Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.} }
Endnote
%0 Conference Paper %T Tracking Representation Dynamics in Large Language Models with Persistent Homology %A Naman Malhotra %A Jay Ambadkar %A Abhinav Gupta %A Kushal Kasivel %A Abbas Schwarz %A Kamillo Ferry %A Anthea Monod %B Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026) %C Proceedings of Machine Learning Research %D 2026 %E Eddie Berman %E Guillermo Bernárdez %E Samantha Chen %E Alex Cloninger %E Timothy Doster %E Tegan Emerson %E J. Elisenda Grigsby %E Henry Kvinge %E Hannah Lawrence %E Tim Marrinan %E Audun Myers %E Mathilde Papillon %E Behrooz Tahmasebi %E Lev Telyatnikov %E Robin Walters %E Melanie Weber %E YuQing Xie %E Eric Yeats %F pmlr-v334-malhotra26a %I PMLR %P 211--245 %U https://proceedings.mlr.press/v334/malhotra26a.html %V 334 %N 2 %X Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.
APA
Malhotra, N., Ambadkar, J., Gupta, A., Kasivel, K., Schwarz, A., Ferry, K. & Monod, A.. (2026). Tracking Representation Dynamics in Large Language Models with Persistent Homology. Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026), in Proceedings of Machine Learning Research 334(2):211-245 Available from https://proceedings.mlr.press/v334/malhotra26a.html.

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