E$(n)$-Equivariant Spherical Decision Surfaces

Pavlo Melnyk, Michael Felsberg, Kostas Daniilidis
Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, PMLR 326:416-432, 2026.

Abstract

We present a constructive derivation of exactly E(n)-equivariant spherical decision surfaces by extending prior O(n)-equivariant hypersphere neurons to include translations. To achieve this, we present a decomposition of the features of the O(n)-equivariant neurons and provide explicit representations for translation and E(n)-transformations to fulfil the respective equivariance constraints. The resulting decision surfaces are exactly E(n)-equivariant without input centring or explicit pairwise differences, and admit explicit closed-form matrix representations. In addition, we numerically verify the correctness of the derivations and perform a downstream check of the resulting geometric primitives.

Cite this Paper

BibTeX
@InProceedings{pmlr-v326-melnyk26a, title = {E$(n)$-{E}quivariant {S}pherical {D}ecision {S}urfaces}, author = {Melnyk, Pavlo and Felsberg, Michael and Daniilidis, Kostas}, booktitle = {Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling}, pages = {416--432}, year = {2026}, editor = {Pouplin, Alison and Vadgama, Sharvaree and Bekkers, Erik and Kaba, Sékou-Oumar and Lawrence, Hannah and Lecha, Manuel and Baker, Elizabeth and Suk, Julian and Walters, Robin and Tomczak, Jakub and Jegelka, Stefanie}, volume = {326}, series = {Proceedings of Machine Learning Research}, month = {26 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v326/main/assets/melnyk26a/melnyk26a.pdf}, url = {https://proceedings.mlr.press/v326/melnyk26a.html}, abstract = {We present a constructive derivation of exactly E(n)-equivariant spherical decision surfaces by extending prior O(n)-equivariant hypersphere neurons to include translations. To achieve this, we present a decomposition of the features of the O(n)-equivariant neurons and provide explicit representations for translation and E(n)-transformations to fulfil the respective equivariance constraints. The resulting decision surfaces are exactly E(n)-equivariant without input centring or explicit pairwise differences, and admit explicit closed-form matrix representations. In addition, we numerically verify the correctness of the derivations and perform a downstream check of the resulting geometric primitives.} }
Endnote
%0 Conference Paper %T E$(n)$-Equivariant Spherical Decision Surfaces %A Pavlo Melnyk %A Michael Felsberg %A Kostas Daniilidis %B Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling %C Proceedings of Machine Learning Research %D 2026 %E Alison Pouplin %E Sharvaree Vadgama %E Erik Bekkers %E Sékou-Oumar Kaba %E Hannah Lawrence %E Manuel Lecha %E Elizabeth Baker %E Julian Suk %E Robin Walters %E Jakub Tomczak %E Stefanie Jegelka %F pmlr-v326-melnyk26a %I PMLR %P 416--432 %U https://proceedings.mlr.press/v326/melnyk26a.html %V 326 %X We present a constructive derivation of exactly E(n)-equivariant spherical decision surfaces by extending prior O(n)-equivariant hypersphere neurons to include translations. To achieve this, we present a decomposition of the features of the O(n)-equivariant neurons and provide explicit representations for translation and E(n)-transformations to fulfil the respective equivariance constraints. The resulting decision surfaces are exactly E(n)-equivariant without input centring or explicit pairwise differences, and admit explicit closed-form matrix representations. In addition, we numerically verify the correctness of the derivations and perform a downstream check of the resulting geometric primitives.
APA
Melnyk, P., Felsberg, M. & Daniilidis, K.. (2026). E$(n)$-Equivariant Spherical Decision Surfaces. Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, in Proceedings of Machine Learning Research 326:416-432 Available from https://proceedings.mlr.press/v326/melnyk26a.html.

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