O$n$ Learning Deep O($n$)-Equivariant Hyperspheres

Pavlo Melnyk, Michael Felsberg, Mårten Wadenbäck, Andreas Robinson, Cuong Le
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:35324-35339, 2024.

Abstract

In this paper, we utilize hyperspheres and regular $n$-simplexes and propose an approach to learning deep features equivariant under the transformations of $n$D reflections and rotations, encompassed by the powerful group of O$(n)$. Namely, we propose O$(n)$-equivariant neurons with spherical decision surfaces that generalize to any dimension $n$, which we call Deep Equivariant Hyperspheres. We demonstrate how to combine them in a network that directly operates on the basis of the input points and propose an invariant operator based on the relation between two points and a sphere, which as we show, turns out to be a Gram matrix. Using synthetic and real-world data in $n$D, we experimentally verify our theoretical contributions and find that our approach is superior to the competing methods for O$(n)$-equivariant benchmark datasets (classification and regression), demonstrating a favorable speed/performance trade-off. The code is available on GitHub.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-melnyk24a, title = {O$n$ Learning Deep O($n$)-Equivariant Hyperspheres}, author = {Melnyk, Pavlo and Felsberg, Michael and Wadenb\"{a}ck, M{\aa}rten and Robinson, Andreas and Le, Cuong}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {35324--35339}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/melnyk24a/melnyk24a.pdf}, url = {https://proceedings.mlr.press/v235/melnyk24a.html}, abstract = {In this paper, we utilize hyperspheres and regular $n$-simplexes and propose an approach to learning deep features equivariant under the transformations of $n$D reflections and rotations, encompassed by the powerful group of O$(n)$. Namely, we propose O$(n)$-equivariant neurons with spherical decision surfaces that generalize to any dimension $n$, which we call Deep Equivariant Hyperspheres. We demonstrate how to combine them in a network that directly operates on the basis of the input points and propose an invariant operator based on the relation between two points and a sphere, which as we show, turns out to be a Gram matrix. Using synthetic and real-world data in $n$D, we experimentally verify our theoretical contributions and find that our approach is superior to the competing methods for O$(n)$-equivariant benchmark datasets (classification and regression), demonstrating a favorable speed/performance trade-off. The code is available on GitHub.} }
Endnote
%0 Conference Paper %T O$n$ Learning Deep O($n$)-Equivariant Hyperspheres %A Pavlo Melnyk %A Michael Felsberg %A Mårten Wadenbäck %A Andreas Robinson %A Cuong Le %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-melnyk24a %I PMLR %P 35324--35339 %U https://proceedings.mlr.press/v235/melnyk24a.html %V 235 %X In this paper, we utilize hyperspheres and regular $n$-simplexes and propose an approach to learning deep features equivariant under the transformations of $n$D reflections and rotations, encompassed by the powerful group of O$(n)$. Namely, we propose O$(n)$-equivariant neurons with spherical decision surfaces that generalize to any dimension $n$, which we call Deep Equivariant Hyperspheres. We demonstrate how to combine them in a network that directly operates on the basis of the input points and propose an invariant operator based on the relation between two points and a sphere, which as we show, turns out to be a Gram matrix. Using synthetic and real-world data in $n$D, we experimentally verify our theoretical contributions and find that our approach is superior to the competing methods for O$(n)$-equivariant benchmark datasets (classification and regression), demonstrating a favorable speed/performance trade-off. The code is available on GitHub.
APA
Melnyk, P., Felsberg, M., Wadenbäck, M., Robinson, A. & Le, C.. (2024). O$n$ Learning Deep O($n$)-Equivariant Hyperspheres. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:35324-35339 Available from https://proceedings.mlr.press/v235/melnyk24a.html.

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