Geometric deep learning and equivariant neural networks
Journal article, 2023

We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning.

Author

Jan E. Gerken

Chalmers, Mathematical Sciences, Algebra and geometry

Berlin Institute for the Foundations of Learning and Data (BIFOLD)

Technische Universität Berlin

Jimmy Aronsson

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

Oscar Carlsson

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

Hampus Linander

University of Gothenburg

Fredrik Ohlsson

Umeå University

Christoffer Petersson

University of Gothenburg

Zenseact AB

Chalmers, Mathematical Sciences, Algebra and geometry

Daniel Persson

Chalmers, Mathematical Sciences, Algebra and geometry

University of Gothenburg

Artificial Intelligence Review

0269-2821 (ISSN) 1573-7462 (eISSN)

Vol. 56 12 14605-14662

Subject Categories

Geometry

Bioinformatics (Computational Biology)

Probability Theory and Statistics

Mathematical Analysis

DOI

10.1007/s10462-023-10502-7

More information

Latest update

7/4/2024 1