Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks
This provides theoretical insights into GNN expressivity for researchers in graph machine learning, but it is incremental as it builds on existing measures for neural networks.
The paper tackles the problem of characterizing the expressiveness of graph convolutional networks (GCNs) by deriving bounds on the number of linear regions, showing that multi-layer GCNs have exponentially more linear regions per parameter than one-layer GCNs, which suggests deeper networks are more expressive.
The research for characterizing GNN expressiveness attracts much attention as graph neural networks achieve a champion in the last five years. The number of linear regions has been considered a good measure for the expressivity of neural networks with piecewise linear activation. In this paper, we present some estimates for the number of linear regions of the classic graph convolutional networks (GCNs) with one layer and multiple-layer scenarios. In particular, we obtain an optimal upper bound for the maximum number of linear regions for one-layer GCNs, and the upper and lower bounds for multi-layer GCNs. The simulated estimate shows that the true maximum number of linear regions is possibly closer to our estimated lower bound. These results imply that the number of linear regions of multi-layer GCNs is exponentially greater than one-layer GCNs per parameter in general. This suggests that deeper GCNs have more expressivity than shallow GCNs.