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.
Bibliographical noteKAUST Repository Item: Exported on 2021-06-30
Acknowledged KAUST grant number(s): OSR-2015-Sensors-2700
Acknowledgements: ENTE. Isufi’s research is supported in part by the KAUST-MIT-TUDCaltech consortium grant OSR-2015-Sensors-2700 Ext. 2018. G.
Kutyniok acknowledges partial support by the Bundesministerium fur Bildung und Forschung (BMBF) through the Berliner Zentrum ¨
for Machine Learning (BZML), Project AP4, by the Deutsche Forschungsgemeinschaft (DFG) through Grants CRC 1114 “Scaling
Cascades in Complex Systems”, Project B07, CRC/TR 109 “Discretization in Geometry and Dynamics”, Projects C02 and C03,
RTG DAEDALUS (RTG 2433), Projects P1 and P3, RTG BIOQIC (RTG 2260), Projects P4 and P9, SPP 1798 “Compressed Sensing in
Information Processing”, Project Coordination and Project Massive MIMO-I/II, by the Berlin Mathematics Research Center MATH+,
Projects EF1-1 and EF1-4, and by the Einstein Foundation Be
This publication acknowledges KAUST support, but has no KAUST affiliated authors.