An elementary method to compute equivariant convolutional kernels on homogeneous spaces for geometric deep learning

03/13/2023

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by   Vincent Knibbeler, et al.

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We develop an elementary method to compute spaces of equivariant maps from a homogeneous space of a Lie group to a module of this group. The Lie group is not required to be compact. More generally we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. This latter case has a natural global algebra structure. We classify the resulting automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.