Graphon Pooling in Graph Neural Networks

Abstract

Graph neural networks (GNNs) have been used effectively in different applications involving the processing of signals on irregular structures modeled by graphs. Relying on the use of shift-invariant graph filters, GNNs extend the operation of convolution to graphs. However, the operations of pooling and sampling are still not clearly defined and the approaches proposed in the literature either modify the graph structure in a way that does not preserve its spectral properties, or require defining a policy for selecting which nodes to keep. In this work, we propose a new strategy for pooling and sampling on GNNs using graphons which preserves the spectral properties of the graph. To do so, we consider the graph layers in a GNN as elements of a sequence of graphs that converge to a graphon. In this way we have no ambiguity in the node labeling when mapping signals from one layer to the other and a spectral representation that is consistent throughout the layers. We evaluate this strategy in a synthetic and a real-world numerical experiment where we show that graphon pooling GNNs are less prone to overfitting and improve upon other pooling techniques, especially when the dimensionality reduction ratios between layers is large.

Convolutional Filtering and Neural Networks with Non Commutative Algebras

“…In (2) we have a generalized representation of the convolution operation between a filter and a signal. Particular instantiations of (2) lead to the traditional signal processing models of time signals and images, and more sophisticated models such as graph signal processing (GSP), graphon signal processing (WSP) among others [7], [8], [19]- [22]. Our goal in this work is to generalize convolutional neural networks to non commutative scenarios and to study their stability properties (Section V).…”

“…Let (A, M, ρ) be a non commutative ASM, where A has generators g 1 , g 2 and let (A, M, ρ) be a perturbed version of (A, M, ρ) associated with the perturbation model in eqn. (19). If the spectral representation of p is in A L0 ∩ A L1 , then…”

“…Let (A, M, ρ) be a non commutative ASM, where A has generators g 1 , g 2 and let (A, M, ρ) be a perturbed version of (A, M, ρ) associated to the perturbation model in eqn. (19). If the spectral representation of p is in A L0 ∩ A L1 , the operator p(S 1 , S 2 ) = ρ(p(g 1 , g 2 )) is stable in the sense of definition 12 with C 0 = 2δL 0 and C 1 = 2δL 1 .…”

“…Let (A, M, ρ) a perturbed version of (A, M, ρ) by means of the perturbation model in eqn. (19). If the spectral representation of p is in A L0 ∩ A L1 , then…”

“…Let (A, M, ρ) be a perturbed version of (A, M, ρ) associated to the perturbation model in eqn. (19).…”