Transferability of graph neural networks: An extended graphon approach
We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important...
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| Vydáno v: | Applied and computational harmonic analysis Ročník 63; s. 48 - 83 |
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| Jazyk: | angličtina |
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01.03.2023
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| ISSN: | 1063-5203, 1096-603X |
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| Abstract | We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability: the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set.
In this paper, we consider a model of transferability based on graphon analysis. Graphons are limit objects of graphs, and, in the graph paradigm, two graphs represent the same phenomenon if both approximate the same graphon. Our main contributions can be summarized as follows: 1) we prove that any fixed GCNN with continuous filters is transferable under graphs that approximate the same graphon, 2) we prove transferability for graphs that approximate unbounded graphon shift operators, which are defined in this paper, and 3) we obtain non-asymptotic approximation results, proving linear stability of GCNNs. This extends current state-of-the-art results which show asymptotic transferability for polynomial filters under graphs that approximate bounded graphons.
•We study the generalization error of graph convolutional neural networks.•Graphs representing the same phenomenon are modeled via graphon analysis.•We show that networks can be transferred between graphs sampling the same phenomenon.•Our analysis allows working with generic continuous filters.•By introducing unbounded graphons, Euclidean CNNs are a special case of our analysis. |
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| AbstractList | We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability: the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set.
In this paper, we consider a model of transferability based on graphon analysis. Graphons are limit objects of graphs, and, in the graph paradigm, two graphs represent the same phenomenon if both approximate the same graphon. Our main contributions can be summarized as follows: 1) we prove that any fixed GCNN with continuous filters is transferable under graphs that approximate the same graphon, 2) we prove transferability for graphs that approximate unbounded graphon shift operators, which are defined in this paper, and 3) we obtain non-asymptotic approximation results, proving linear stability of GCNNs. This extends current state-of-the-art results which show asymptotic transferability for polynomial filters under graphs that approximate bounded graphons.
•We study the generalization error of graph convolutional neural networks.•Graphs representing the same phenomenon are modeled via graphon analysis.•We show that networks can be transferred between graphs sampling the same phenomenon.•Our analysis allows working with generic continuous filters.•By introducing unbounded graphons, Euclidean CNNs are a special case of our analysis. |
| Author | Kutyniok, Gitta Maskey, Sohir Levie, Ron |
| Author_xml | – sequence: 1 givenname: Sohir orcidid: 0000-0002-9691-6712 surname: Maskey fullname: Maskey, Sohir email: maskey@math.lmu.de organization: Department of Mathematics, LMU Munich, 80333 Munich, Germany – sequence: 2 givenname: Ron surname: Levie fullname: Levie, Ron email: levieron@technion.ac.il organization: Faculty of Mathematics, Technion - Israel Institute of Technology, Israel – sequence: 3 givenname: Gitta orcidid: 0000-0001-9738-2487 surname: Kutyniok fullname: Kutyniok, Gitta email: kutyniok@math.lmu.de organization: Department of Mathematics, LMU Munich, 80333 Munich, Germany |
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| Keywords | Graphon Stability Graph neural network 68T07 Transferability 68R10 Generalization 47A60 Spectral methods |
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