On the Expressive Power of Permutation-Equivariant Weight-Space Networks

Adir Dayan, Yam Eitan, Haggai Maron
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:23179-23249, 2026.

Abstract

Weight-space learning studies neural architectures that operate directly on the parameters of other neural networks. Motivated by the growing availability of pretrained models, recent work has demonstrated the effectiveness of weight-space networks across a wide range of tasks. SOTA weight-space networks rely on permutation-equivariant designs to improve generalization. However, this may negatively affect expressive power, warranting theoretical investigation. Importantly, unlike other structured domains, weight-space learning targets maps operating on both weight and function spaces, making expressivity analysis particularly subtle. While a few prior works provide partial expressivity results, a comprehensive characterization is still missing. In this work, we address this gap by developing a systematic theory for expressivity of weight-space networks. We first prove that all prominent permutation-equivariant networks are equivalent in expressive power. We then establish universality in both weight- and function-space settings under mild, natural assumptions on the input weights, and characterize the edge-case regimes where universality no longer holds. Guided by our theoretical results, we show that slight modifications to existing weight-space models yield a 34% improvement over prior SOTA, demonstrating the practical relevance of our framework.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-dayan26a, title = {On the Expressive Power of Permutation-Equivariant Weight-Space Networks}, author = {Dayan, Adir and Eitan, Yam and Maron, Haggai}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {23179--23249}, 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/dayan26a/dayan26a.pdf}, url = {https://proceedings.mlr.press/v306/dayan26a.html}, abstract = {Weight-space learning studies neural architectures that operate directly on the parameters of other neural networks. Motivated by the growing availability of pretrained models, recent work has demonstrated the effectiveness of weight-space networks across a wide range of tasks. SOTA weight-space networks rely on permutation-equivariant designs to improve generalization. However, this may negatively affect expressive power, warranting theoretical investigation. Importantly, unlike other structured domains, weight-space learning targets maps operating on both weight and function spaces, making expressivity analysis particularly subtle. While a few prior works provide partial expressivity results, a comprehensive characterization is still missing. In this work, we address this gap by developing a systematic theory for expressivity of weight-space networks. We first prove that all prominent permutation-equivariant networks are equivalent in expressive power. We then establish universality in both weight- and function-space settings under mild, natural assumptions on the input weights, and characterize the edge-case regimes where universality no longer holds. Guided by our theoretical results, we show that slight modifications to existing weight-space models yield a 34% improvement over prior SOTA, demonstrating the practical relevance of our framework.} }
Endnote
%0 Conference Paper %T On the Expressive Power of Permutation-Equivariant Weight-Space Networks %A Adir Dayan %A Yam Eitan %A Haggai Maron %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-dayan26a %I PMLR %P 23179--23249 %U https://proceedings.mlr.press/v306/dayan26a.html %V 306 %X Weight-space learning studies neural architectures that operate directly on the parameters of other neural networks. Motivated by the growing availability of pretrained models, recent work has demonstrated the effectiveness of weight-space networks across a wide range of tasks. SOTA weight-space networks rely on permutation-equivariant designs to improve generalization. However, this may negatively affect expressive power, warranting theoretical investigation. Importantly, unlike other structured domains, weight-space learning targets maps operating on both weight and function spaces, making expressivity analysis particularly subtle. While a few prior works provide partial expressivity results, a comprehensive characterization is still missing. In this work, we address this gap by developing a systematic theory for expressivity of weight-space networks. We first prove that all prominent permutation-equivariant networks are equivalent in expressive power. We then establish universality in both weight- and function-space settings under mild, natural assumptions on the input weights, and characterize the edge-case regimes where universality no longer holds. Guided by our theoretical results, we show that slight modifications to existing weight-space models yield a 34% improvement over prior SOTA, demonstrating the practical relevance of our framework.
APA
Dayan, A., Eitan, Y. & Maron, H.. (2026). On the Expressive Power of Permutation-Equivariant Weight-Space Networks. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:23179-23249 Available from https://proceedings.mlr.press/v306/dayan26a.html.

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