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Embed Me If You Can: A Geometric Perceptron
Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-6091-861X
Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-6096-3648
Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-0675-2794
2021 (English)In: Proceedings 2021 IEEE/CVF International Conference on Computer Vision ICCV 2021, Institute of Electrical and Electronics Engineers (IEEE), 2021, p. 1256-1264Conference paper, Published paper (Refereed)
Abstract [en]

Solving geometric tasks involving point clouds by using machine learning is a challenging problem. Standard feed-forward neural networks combine linear or, if the bias parameter is included, affine layers and activation functions. Their geometric modeling is limited, which motivated the prior work introducing the multilayer hypersphere perceptron (MLHP). Its constituent part, i.e., the hypersphere neuron, is obtained by applying a conformal embedding of Euclidean space. By virtue of Clifford algebra, it can be implemented as the Cartesian dot product of inputs and weights. If the embedding is applied in a manner consistent with the dimensionality of the input space geometry, the decision surfaces of the model units become combinations of hyperspheres and make the decision-making process geometrically interpretable for humans. Our extension of the MLHP model, the multilayer geometric perceptron (MLGP), and its respective layer units, i.e., geometric neurons, are consistent with the 3D geometry and provide a geometric handle of the learned coefficients. In particular, the geometric neuron activations are isometric in 3D, which is necessary for rotation and translation equivariance. When classifying the 3D Tetris shapes, we quantitatively show that our model requires no activation function in the hidden layers other than the embedding to outperform the vanilla multilayer perceptron. In the presence of noise in the data, our model is also superior to the MLHP.

Place, publisher, year, edition, pages
Institute of Electrical and Electronics Engineers (IEEE), 2021. p. 1256-1264
Series
IEEE International Conference on Computer Vision. Proceedings, ISSN 1550-5499, E-ISSN 2380-7504
National Category
Computer graphics and computer vision
Identifiers
URN: urn:nbn:se:liu:diva-183312DOI: 10.1109/iccv48922.2021.00131ISI: 000797698901044ISBN: 9781665428125 (electronic)ISBN: 9781665428132 (print)OAI: oai:DiVA.org:liu-183312DiVA, id: diva2:1641525
Conference
IEEE/CVF International Conference on Computer Vision (ICCV), 10-17 October 2021 (Virtual Event), Montreal, QC, Canada
Note

Funding: Wallenberg AI, Autonomous Systems and Software Program (WASP); Swedish Research Council [2018-04673]; strategic research environment ELLIIT

Available from: 2022-03-02 Created: 2022-03-02 Last updated: 2025-02-07Bibliographically approved
In thesis
1. Spherical NeurO(n)s for Geometric Deep Learning
Open this publication in new window or tab >>Spherical NeurO(n)s for Geometric Deep Learning
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Felix Klein’s Erlangen Programme of 1872 introduced a methodology to unify non-Euclidean geometries. Similarly, geometric deep learning (GDL) constitutes a unifying framework for various neural network architectures. GDL is built from the first principles of geometry—symmetry and scale separation—and enables tractable learning in high dimensions. Symmetries play a vital role in preserving structural information of geometric data and allow models (i.e., neural networks) to adjust to different geometric transformations. 

In this context, spheres exhibit a maximal set of symmetries compared to other geometric entities in Euclidean space. The orthogonal group O(n) fully encapsulates the symmetry structure of an nD sphere, including both rotational and reflection symmetries. In this thesis, we focus on integrating these symmetries into a model as an inductive bias, which is a crucial requirement for addressing problems in 3D vision as well as in natural sciences and their related applications. 

In Paper A, we focus on 3D geometry and use the symmetries of spheres as geometric entities to construct neurons with spherical decision surfaces—spherical neurons—using a conformal embedding of Euclidean space. We also demonstrate that spherical neuron activations are non-linear due to the inherent non-linearity of the input embedding, and thus, do not necessarily require an activation function. In addition, we show graphically, theoretically, and experimentally that spherical neuron activations are isometries in Euclidean space, which is a prerequisite for the equivariance contributions of our subsequent work. 

In Paper B, we closely examine the isometry property of the spherical neurons in the context of equivariance under 3D rotations (i.e., SO(3)-equivariance). Focusing on 3D in this work and based on a minimal set of four spherical neurons (one learned spherical decision surface and three copies), the centers of which are rotated into the corresponding vertices of a regular tetrahedron, we construct a spherical filter bank. We call it a steerable 3D spherical neuron because, as we verify later, it constitutes a steerable filter. Finally, we derive a 3D steerability constraint for a spherical neuron (i.e., a single spherical decision surface). 

In Paper C, we present a learnable point-cloud descriptor invariant under 3D rotations and reflections, i.e., the O(3) actions, utilizing the steerable 3D spherical neurons we introduced previously, as well as vector neurons from related work. Specifically, we propose an embedding of the 3D steerable neurons into 4D vector neurons, which leverages end-to-end training of the model. The resulting model, termed TetraSphere, sets a new state-of-the-art performance classifying randomly rotated real-world object scans. Thus, our results reveal the practical value of steerable 3D spherical neurons for learning in 3D Euclidean space. 

In Paper D, we generalize to nD the concepts we previously established in 3D, and propose O(n)-equivariant neurons with spherical decision surfaces, which we call Deep Equivariant Hyper-spheres. 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. 

In summary, this thesis introduces techniques based on spherical neurons that enhance the GDL framework, with a specific focus on equivariant and invariant learning on point sets. 

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2024. p. 37
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2393
National Category
Computer graphics and computer vision
Identifiers
urn:nbn:se:liu:diva-207304 (URN)10.3384/9789180756808 (DOI)9789180756792 (ISBN)9789180756808 (ISBN)
Public defence
2024-09-27, Ada Lovelace, B-building, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Note

Funding:  Wallenberg AI, Autonomous Systems and Software Program (WASP); National Academic Infrastructure for Supercomputing in Sweden (NAISS) partially funded by the Swedish Research Council through grant agreement no. 2022-06725, and by the Berzelius resource provided by the Knut and Alice Wallenberg Foundation at the National Supercomputer Centre.

Available from: 2024-09-03 Created: 2024-09-03 Last updated: 2025-02-07Bibliographically approved

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Melnyk, PavloFelsberg, MichaelWadenbäck, Mårten

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