Geometric Deep Learning and Equivariant Neural Networks

Gerken, Jan E.; Aronsson, Jimmy; Carlsson, Oscar; Linander, Hampus; Ohlsson, Fredrik; Petersson, Christoffer; Persson, Daniel

doi:10.48550/arxiv.2105.13926

Cited by 5 publications

(11 citation statements)

References 60 publications

(153 reference statements)

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“…when the domain M is a (possibly curved) manifold. This research field is referred to as geometric deep learning, an umbrella term first coined in [3] (see [2] and [14] for recent reviews). The results of the present paper may thus be viewed as probing a small corner of the vast field of geometric deep learning.…”

Section: Discussionmentioning

confidence: 99%

“…In general, however, the group G can act through non-trivial representations on both the manifold G/H and the feature maps (see e.g. [17,5,1,14]).…”

Section: Group Equivariant Networkmentioning

confidence: 99%

“…where in (A.3) we used the transformation property of the spherical harmonics and in (A.4) that the Wigner matrices form a unitary representation. For a summary of the necessary formulae and the conventions 6 , see, e.g., [14].…”

Section: A Mathematical Properties Of the Final S2cnn Layermentioning

confidence: 99%

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Equivariance versus Augmentation for Spherical Images

Gerken¹,

Carlsson²,

Linander³

et al. 2022

Preprint

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We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems. The equivariant spherical networks used in the experiments will be made available at https: //github.com/JanEGerken/sem_seg_s2cnn.

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Section: Discussionmentioning

confidence: 99%

“…In general, however, the group G can act through non-trivial representations on both the manifold G/H and the feature maps (see e.g. [17,5,1,14]).…”

Section: Group Equivariant Networkmentioning

confidence: 99%

See 1 more Smart Citation

Equivariance versus Augmentation for Spherical Images

Gerken¹,

Carlsson²,

Linander³

et al. 2022

Preprint

Self Cite

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show abstract

“…In particular there has been a focus on describing linear equivariant functions, which can be alternated with non-linearities to obtain equivariant neural network architectures [1, 4,5,12,21]. Recent surveys of the theory include [3,15,41].…”

Section: Related Workmentioning

confidence: 99%

“…Let us call clouds X with i∈[m] x i = 0 and x 0 = 0 centered. Consider the basis (K i ) i∈ [15] of L 0 (2, 1) described in Section C. All of their action on elements of the form X ⊗ X (see in particular the final paragraph of the mentioned section) are identically zero, except for…”

Section: A7 the Two-cloud Architecturementioning

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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Prediction of molecular field points using SE(3)-transformer model

Florian

Mahmoud

Lill

2023

Mach. Learn.: Sci. Technol.

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Due to their computational eﬃciency, 2D ﬁngerprints are typically used in similarity-based high-content screening. The interaction of a ligand with its target protein, however, relies on its physicochemical interactions in 3D space. Thus, ligands with diﬀerent 2D scaﬀolds can bind to the same protein if these ligands share similar interaction patterns. Molecular ﬁelds can represent those interaction proﬁles. For eﬃciency, the extrema of those molecular ﬁelds, named ﬁeld points, are used to quantify the ligand similarity in 3D. The calculation of ﬁeld points involves the evaluation of the interaction energy between the ligand and a small probe shifted on a ﬁne grid representing the molecular surface. These calculations are computationally prohibitive for large datasets of ligands, making ﬁeld point representations of molecules intractable for high-content screening. Here, we overcome this roadblock by one-shot prediction of ﬁeld points using generative neural networks based on the molecular structure alone. Field points are predicted by training an SE(3)-Transformer, an equivariant, attention-based graph neural network architecture, on a large set of ligands with ﬁeld point data. Resulting data demonstrates the feasibility of this approach to precisely generate negative, positive and hydrophobic ﬁeld points within 0.5 Å of the ground truth for a diverse set of drug-like molecules.

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Geometric Deep Learning and Equivariant Neural Networks

Cited by 5 publications

References 60 publications

Equivariance versus Augmentation for Spherical Images

Equivariance versus Augmentation for Spherical Images

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

Prediction of molecular field points using SE(3)-transformer model

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