Embed Me If You Can: A Geometric Perceptron

Melnyk, Pavlo; Felsberg, Michael; Wadenbäck, Mårten

doi:10.1109/iccv48922.2021.00131

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…In the domain of 3D CNNs, two of the possible data representations are point clouds and volumetric data. Many approaches that seek equivariance to space rotations are for CNNs that process point cloud data (Thomas et al, 2018;Chen et al, 2021;Melnyk et al, 2021;Thomas, 2020).…”

Section: Related Workmentioning

confidence: 99%

Moving Frame Net: SE(3)-Equivariant Network for Volumes

Sangalli¹,

Blusseau²,

Velasco-Forero³

et al. 2022

Preprint

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Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The method of moving frames is classical for deriving operators invariant to the action of a Lie group in a manifold. Recently, a rotation and translation equivariant neural network for image data was proposed based on the moving frames approach. In this paper we significantly improve that approach by reducing the computation of moving frames to only one, at the input stage, instead of repeated computations at each layer. The equivariance of the resulting architecture is proved theoretically and we build a rotation and translation equivariant neural network to process volumes, i.e. signals on the 3D space. Our trained model overperforms the benchmarks in the medical volume classification of most of the tested datasets from MedMNIST3D.

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Section: Related Workmentioning

confidence: 99%

Moving Frame Net: SE(3)-Equivariant Network for Volumes

Sangalli¹,

Blusseau²,

Velasco-Forero³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…There exist a number of high-performing deep learning architectures for 3D point cloud processing, mostly targeted for recognition, classification and segmentation, including methods that do not take rotation equivariance into account [31,51] and methods that do consider the effects of rotations [7,13,29,34]. The approach most similar in spirit to ours is [44], but while we let every point in the point cloud gather information from all others to obtain rotation invariant and permutation equivariant features, they use the sorted Gram matrix of local neighbourhoods to obtain local rotation and permutation invariant features.…”

Section: Related Workmentioning

confidence: 99%

ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds

Bökman¹,

Kahl²,

Flinth³

2021

Preprint

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In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries.We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of essential matrices in stereo vision. References

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“…The approach described herein applies geometric algebra to train deep learning models on point clouds, but using geometric algebra (also known as Clifford algebra) to structure the operations of neural networks is not a new concept. Prior work has introduced architectures which generate multivectors using the mathematical structure of geometric algebra as an extension of complex numbers (Pearson & Bisset, 1992;1994) and directly for geometric applications (Bayro-Corrochano et al, 1996;Buchholz & Sommer, 2008;Melnyk et al, 2020).…”

Section: Related Workmentioning

confidence: 99%

Geometric Algebra Attention Networks for Small Point Clouds

Spellings¹

2021

Preprint

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Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate-a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two-or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation-and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

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Embed Me If You Can: A Geometric Perceptron

Cited by 5 publications

References 14 publications

Moving Frame Net: SE(3)-Equivariant Network for Volumes

Moving Frame Net: SE(3)-Equivariant Network for Volumes

ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds

Geometric Algebra Attention Networks for Small Point Clouds

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