Homogeneous vector bundles and G-equivariant convolutional neural networks
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G-equivariant convolutional neural networks (GCNNs) is a geometric deep learning model for data defined on a homogeneous G-space ℳ. GCNNs are designed to respect the global symmetry in ℳ, thereby facilitating learning. In this paper, we analyze GCNNs on homogeneous spaces ℳ = G/K in the case of unimodular Lie groups G and compact subgroups K ≤ G. We demonstrate that homogeneous vector bundles is the natural setting for GCNNs. We also use reproducing kernel Hilbert spaces to obtain a precise criterion for expressing G-equivariant layers as convolutional layers. This criterion is then rephrased as a bandwidth criterion, leading to even stronger results for some groups.
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