Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphs

In this article we propose a pooling approach for convolutional information processing on graphs relying on the theory of graphons and limits of dense graph sequences. We present three methods that exploit the induced graphon representation of graphs and graph signals on partitions of
$\mathbf{[0,1]^{2}}$
in the graphon space. As a result we derive low dimensional representations of the convolutional operators, while a dimensionality reduction of the signals is achieved by simple local interpolation of functions in
$\boldsymbol{L^{2}}\mathbf{([0,1])}$
. We prove that those low dimensional representations constitute a convergent sequence of graphs and graph signals, respectively. The methods proposed and the theoretical guarantees that we provide show that the reduced graphs and signals inherit spectral-structural properties of the original quantities. We evaluate our approach with a set of numerical experiments performed on graph neural networks (GNNs) that rely on graphon pooling. We observe that graphon pooling performs significantly better than other approaches proposed in the literature when dimensionality reduction ratios between layers are large. We also observe that when graphon pooling is used we have, in general, less overfitting and lower computational cost.