# On the Transferability of Spectral Graph Filters

## TL;DR

This paper demonstrates that spectral graph filters can be stable and transferable across different graphs by introducing the Cayley smoothness space, challenging the misconception that spectral filters lack stability.

## Contribution

The paper introduces the Cayley smoothness space, proving spectral filters within it are linearly stable and transferable across graphs, advancing understanding of spectral filter generalization.

## Key findings

- Spectral graph filters in the Cayley smoothness space are linearly stable.
- Filters in this space can approximate any generic spectral filter.
- Spectral filters are transferable due to stability and equivariance.

## Abstract

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.