Learning Deep O($n$)-Equivariant Hyperspheres

Pavlo Melnyk, Michael Felsberg, Mårten Wadenbäck, Andreas Robinson, Cuong Le

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Keywords: spherical neurons, rotational equivariance, regular simplexes, geometric deep learning

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TL;DR: O(n)-equivariance via learned spherical decision surfaces and regular n-simplexes

Abstract: This paper presents an approach to learning (deep) $n$D features equivariant under orthogonal transformations, utilizing hyperspheres and regular $n$-simplexes. Our main contributions are theoretical and tackle major challenges in geometric deep learning such as equivariance and invariance under geometric transformations. Namely, we enrich the recently developed theory of steerable 3D spherical neurons---$\textup{SO}(3)$-equivariant filter banks based on neurons with spherical decision surfaces---by extending said neurons to $n$D, which we call deep equivariant hyperspheres, and enabling their multi-layer construction. Using synthetic and real-world data in $n$D, we experimentally verify our theoretical contributions and find that our approach is superior to the baselines for small training data sets in all but one case.

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Submission Number: 5920