Geometric deep learning and equivariant neural networks
Artikel i vetenskaplig tidskrift, 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.
Författare
Jan E. Gerken
Berlin Institute for the Foundations of Learning and Data (BIFOLD)
Technische Universität Berlin
Jimmy Aronsson
Chalmers, Matematiska vetenskaper, Algebra och geometri
Göteborgs universitet
Oscar Carlsson
Göteborgs universitet
Chalmers, Matematiska vetenskaper, Algebra och geometri
Hampus Linander
Göteborgs universitet
Fredrik Ohlsson
Umeå universitet
Christoffer Petersson
Chalmers, Matematiska vetenskaper, Algebra och geometri
Zenseact AB
Göteborgs universitet
Daniel Persson
Göteborgs universitet
Chalmers, Matematiska vetenskaper, Algebra och geometri
Artificial Intelligence Review
0269-2821 (ISSN) 1573-7462 (eISSN)
Vol. 56 12 14605-14662Ämneskategorier
Geometri
Matematisk analys
DOI
10.1007/s10462-023-10502-7