O$n$ Learning Deep O($n$)-Equivariant Hyperspheres
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911892990066688 |
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| author | Melnyk, Pavlo Felsberg, Michael Wadenbäck, Mårten Robinson, Andreas Le, Cuong |
| author_facet | Melnyk, Pavlo Felsberg, Michael Wadenbäck, Mårten Robinson, Andreas Le, Cuong |
| contents | 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 at https://github.com/pavlo-melnyk/equivariant-hyperspheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_15613 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | O$n$ Learning Deep O($n$)-Equivariant Hyperspheres Melnyk, Pavlo Felsberg, Michael Wadenbäck, Mårten Robinson, Andreas Le, Cuong Machine Learning 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 at https://github.com/pavlo-melnyk/equivariant-hyperspheres. |
| title | O$n$ Learning Deep O($n$)-Equivariant Hyperspheres |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2305.15613 |