  This paper focuses on spectral filters on graphs, namely filters defined as
elementwise multiplication in the frequency domain of a graph. In many graph
signal processing settings, it is important to transfer a filter from one graph
to another. One example is in graph convolutional neural networks (ConvNets),
where the dataset consists of signals defined on many different graphs, and the
learned filters should generalize to signals on new graphs, not present in the
training set. A necessary condition for transferability (the ability to
transfer filters) is stability. Namely, given a graph filter, if we add a small
perturbation to the graph, then the filter on the perturbed graph is a small
perturbation of the original filter. It is a common misconception that spectral
filters are not stable, and this paper aims at debunking this mistake. We
introduce a space of filters, called the Cayley smoothness space, that contains
the filters of state-of-the-art spectral filtering methods, and whose filters
can approximate any generic spectral filter. For filters in this space, the
perturbation in the filter is bounded by a constant times the perturbation in
the graph, and filters in the Cayley smoothness space are thus termed linearly
stable. By combining stability with the known property of equivariance, we
prove that graph spectral filters are transferable.
