  This paper focuses on spectral graph convolutional neural networks
(ConvNets), where filters are defined as elementwise multiplication in the
frequency domain of a graph. In machine learning settings where the dataset
consists of signals defined on many different graphs, the trained ConvNet
should generalize to signals on graphs unseen in the training set. It is thus
important to transfer ConvNets between graphs. Transferability, which is a
certain type of generalization capability, can be loosely defined as follows:
if two graphs describe the same phenomenon, then a single filter or ConvNet
should have similar repercussions on both graphs. This paper aims at debunking
the common misconception that spectral filters are not transferable. We show
that if two graphs discretize the same "continuous" space, then a spectral
filter or ConvNet has approximately the same repercussion on both graphs. Our
analysis is more permissive than the standard analysis. Transferability is
typically described as the robustness of the filter to small graph
perturbations and re-indexing of the vertices. Our analysis accounts also for
large graph perturbations. We prove transferability between graphs that can
have completely different dimensions and topologies, only requiring that both
graphs discretize the same underlying space in some generic sense.
