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
Berlin Institute for the Foundations of Learning and Data (BIFOLD)
Technische Universität Berlin
Jimmy Aronsson
Chalmers, Mathematical Sciences, Algebra and geometry
University of Gothenburg
Oscar Carlsson
University of Gothenburg
Chalmers, Mathematical Sciences, Algebra and geometry
Hampus Linander
University of Gothenburg
Fredrik Ohlsson
Umeå University
Christoffer Petersson
Chalmers, Mathematical Sciences, Algebra and geometry
Zenseact AB
University of Gothenburg
Daniel Persson
University of Gothenburg
Chalmers, Mathematical Sciences, Algebra and geometry
Artificial Intelligence Review
0269-2821 (ISSN) 1573-7462 (eISSN)
Vol. 56 12 14605-14662Subject Categories
Geometry
Mathematical Analysis
DOI
10.1007/s10462-023-10502-7