Harmformer: Harmonic Networks Meet Transformers for Continuous Roto-Translation Equivariance

Tomáš Karella, Adam Harmanec, Jan Kotera, Jan Blažek, Filip Šroubek
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:884-919, 2026.

Abstract

Convolutional Neural Networks exhibit inherent equivariance to image translation, leading to efficient parameter and data usage, faster learning, and improved robustness. The concept of translation equivariant networks has been successfully extended to rotation transformation using group convolution for discrete rotation groups and harmonic functions for the continuous rotation group encompassing $360^\circ$. We explore the compatibility of the Self-Attention mechanism with full rotation equivariance, in contrast to previous studies that focused on discrete rotation. We introduce the Harmformer, a harmonic transformer with a convolutional stem that achieves equivariance for both translation and continuous rotation. Accompanied by an end-to-end equivariance proof, the Harmformer not only outperforms previous equivariant transformers, but also demonstrates inherent stability under any continuous rotation, even without seeing rotated samples during training.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-karella26a, title = {Harmformer: Harmonic Networks Meet Transformers for Continuous Roto-Translation Equivariance}, author = {Karella, Tom\'a\v{s} and Harmanec, Adam and Kotera, Jan and Bla\v{z}ek, Jan and \v{S}roubek, Filip}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {884--919}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/karella26a/karella26a.pdf}, url = {https://proceedings.mlr.press/v282/karella26a.html}, abstract = {Convolutional Neural Networks exhibit inherent equivariance to image translation, leading to efficient parameter and data usage, faster learning, and improved robustness. The concept of translation equivariant networks has been successfully extended to rotation transformation using group convolution for discrete rotation groups and harmonic functions for the continuous rotation group encompassing $360^\circ$. We explore the compatibility of the Self-Attention mechanism with full rotation equivariance, in contrast to previous studies that focused on discrete rotation. We introduce the Harmformer, a harmonic transformer with a convolutional stem that achieves equivariance for both translation and continuous rotation. Accompanied by an end-to-end equivariance proof, the Harmformer not only outperforms previous equivariant transformers, but also demonstrates inherent stability under any continuous rotation, even without seeing rotated samples during training.} }
Endnote
%0 Conference Paper %T Harmformer: Harmonic Networks Meet Transformers for Continuous Roto-Translation Equivariance %A Tomáš Karella %A Adam Harmanec %A Jan Kotera %A Jan Blažek %A Filip Šroubek %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-karella26a %I PMLR %P 884--919 %U https://proceedings.mlr.press/v282/karella26a.html %V 282 %X Convolutional Neural Networks exhibit inherent equivariance to image translation, leading to efficient parameter and data usage, faster learning, and improved robustness. The concept of translation equivariant networks has been successfully extended to rotation transformation using group convolution for discrete rotation groups and harmonic functions for the continuous rotation group encompassing $360^\circ$. We explore the compatibility of the Self-Attention mechanism with full rotation equivariance, in contrast to previous studies that focused on discrete rotation. We introduce the Harmformer, a harmonic transformer with a convolutional stem that achieves equivariance for both translation and continuous rotation. Accompanied by an end-to-end equivariance proof, the Harmformer not only outperforms previous equivariant transformers, but also demonstrates inherent stability under any continuous rotation, even without seeing rotated samples during training.
APA
Karella, T., Harmanec, A., Kotera, J., Blažek, J. & Šroubek, F.. (2026). Harmformer: Harmonic Networks Meet Transformers for Continuous Roto-Translation Equivariance. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:884-919 Available from https://proceedings.mlr.press/v282/karella26a.html.

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