Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation

Haruka Ezoe, Hiroki Matsumoto, Ryohei Hisano
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:28492-28527, 2026.

Abstract

Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying node embeddings, whose usefulness depends on well defined stability properties across nodes and across time. We introduce Unfolded Laplacian Spectral Embedding (ULSE), a principled extension of unfolded adjacency spectral embedding to normalized Laplacian operators, a setting where stability guarantees have remained out of reach. We prove that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model. Moreover, the Laplacian formulation yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings. Empirical results on synthetic and real-world dynamic networks validate the theory.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-ezoe26a, title = {Unfolded {L}aplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation}, author = {Ezoe, Haruka and Matsumoto, Hiroki and Hisano, Ryohei}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {28492--28527}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/ezoe26a/ezoe26a.pdf}, url = {https://proceedings.mlr.press/v306/ezoe26a.html}, abstract = {Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying node embeddings, whose usefulness depends on well defined stability properties across nodes and across time. We introduce Unfolded Laplacian Spectral Embedding (ULSE), a principled extension of unfolded adjacency spectral embedding to normalized Laplacian operators, a setting where stability guarantees have remained out of reach. We prove that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model. Moreover, the Laplacian formulation yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings. Empirical results on synthetic and real-world dynamic networks validate the theory.} }
Endnote
%0 Conference Paper %T Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation %A Haruka Ezoe %A Hiroki Matsumoto %A Ryohei Hisano %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-ezoe26a %I PMLR %P 28492--28527 %U https://proceedings.mlr.press/v306/ezoe26a.html %V 306 %X Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying node embeddings, whose usefulness depends on well defined stability properties across nodes and across time. We introduce Unfolded Laplacian Spectral Embedding (ULSE), a principled extension of unfolded adjacency spectral embedding to normalized Laplacian operators, a setting where stability guarantees have remained out of reach. We prove that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model. Moreover, the Laplacian formulation yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings. Empirical results on synthetic and real-world dynamic networks validate the theory.
APA
Ezoe, H., Matsumoto, H. & Hisano, R.. (2026). Unfolded Laplacian Spectral Embedding: A Theoretically Grounded Approach to Dynamic Network Representation. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:28492-28527 Available from https://proceedings.mlr.press/v306/ezoe26a.html.

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