More precisely, we consider networks with a single hidden layer,obtained bysumming channels formed byapplying anequivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linearoutputlayer.

The proof is given by [11]. Eq. (13)is clearlyO(3)-subequivariant, but theO(3)-subequivariant function is unnecessarily the form like Eq. (13). Then there must exit functionss( Z,h) and s ( Z,h), satisfying ˆf( Z,h) = [ Z, g]s( Z,h)+ Z s ( Z,h). Note thatf by Eq. (14) can also be considered as a function of both Z and g, and it is universal accordingtoProposition1. When f reducestoafunctionof Z byfixing g,thenbyTheorem1,itis 4 still universal with respect tothe subgroup that leaves g unchanged.

Inthisworkwestudy theconvergence properties of the gradient descent algorithm when used to train GNNs.

A properly extracted subgraph consists of a small number ofcritical neighbors, while excluding irrelevantones.

GraphNeuralNetworks.jl is an open-source framework for deep learning on graphs, written in the Julia programming language. It supports multiple GPU backends, generic sparse or dense graph representations, and offers convenient interfaces for manipulating standard, heterogeneous, and temporal graphs with attributes at the node, edge, and graph levels. The framework allows users to define custom graph convolutional layers using gather/scatter message-passing primitives or optimized fused operations. It also includes several popular layers, enabling efficient experimentation with complex deep architectures. The package is available on GitHub: https://github.com/JuliaGraphs/