The Graphon Fourier Transform

Ruiz, Luana; Chamon, Luiz F. O.; Ribeiro, Alejandro

doi:10.1109/icassp40776.2020.9053882

Cited by 16 publications

(18 citation statements)

References 26 publications

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“…Theorem 11.53 in [18] gives a formal statement and proof of this fact. The Graphon Fourier Transform has also be defined via the resolvent equation in [10]. Similarly, the degree matrix is also determined by the graphon [28], and thus, the spectrum of the graph Laplacian can also be investigated through the resolvent equation.…”

Section: Fredholm Integral Equations and Graphon Filtersmentioning

confidence: 99%

“…Graph signal processing is an emerging topic which endeavours to explain the evolution of signals supported on an irregular domain, modeled as a graph [1]- [10]. The interconnections between nodes of the graph in some way capture interdependencies of data supported on the nodes.…”

Section: Introductionmentioning

confidence: 99%

“…Other works in the signal processing literature have leveraged the spectral convergence properties of random graphs. For example, in [10] the authors leverage the spectral convergence properties of the scaled adjacency matrix to extend the notion of the graph Fourier transform to graphons, which they dub the Graphon Fourier Transform.…”

Section: Introductionmentioning

confidence: 99%

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Graphon Filters: Graph Signal Processing in the Limit

Morency

Leus

2021

IEEE Trans. Signal Process.

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Graph signal processing is an emerging field which aims to model processes that exist on the nodes of a network and are explained through diffusion over this structure. Graph signal processing works have heretofore assumed knowledge of the graph shift operator. Our approach is to investigate the question of graph filtering on a graph about which we only know a model. To do this we leverage the theory of graphons proposed by L. Lovasz and B. Szegedy. We make three key contributions to the emerging field of graph signal processing. We show first that filters defined over the scaled adjacency matrix of a random graph drawn from a graphon converge to filters defined over the Fredholm integral operator with the graphon as its kernel. Second, leveraging classical findings from the theory of the numerical solution of Fredholm integral equations, we define the Fourier-Galerkin shift operator. Lastly, using the Fourier-Galerkin shift operator, we derive a graph filter design algorithm which only depends on the graphon, and thus depends only on the probabilistic structure of the graph instead of the particular graph itself. The derived graphon filtering algorithm is verified through simulations on a variety of random graph models.

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Section: Fredholm Integral Equations and Graphon Filtersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Graphon Filters: Graph Signal Processing in the Limit

Morency

Leus

2021

IEEE Trans. Signal Process.

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“…Graphon signals are defined as functions X ∈ L2([0, 1]). They can be seen as limits of graph signals supported on graphs converging to a graphon [27], or as generating models for graph signals xn supported on the deterministic graphḠn (5) or on the stochastic graph Gn (6) and given by…”

Section: Graphon Signals and Graphon Convolutionsmentioning

confidence: 99%

Graphon and Graph Neural Network Stability

Ruiz

Wang

Ribeiro

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Graph neural networks (GNNs) are learning architectures that rely on knowledge of the graph structure to generate meaningful representations of large-scale network data. GNN stability is thus important as in real-world scenarios there are typically uncertainties associated with the graph. We analyze GNN stability using kernel objects called graphons. Graphons are both limits of convergent graph sequences and generating models for deterministic and stochastic graphs. Building upon the theory of graphon signal processing, we define graphon neural networks and analyze their stability to graphon perturbations. We then extend this analysis by interpreting the graphon neural network as a generating model for GNNs on deterministic and stochastic graphs instantiated from the original and perturbed graphons. We observe that GNNs are stable to graphon perturbations with a stability bound that decreases asymptotically with the size of the graph. This asymptotic behavior is further demonstrated in an experiment of movie recommendation.

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“…As a significant tool in signal processing community, Fourier Transform (FT) plays important roles in academic and industrial sectors [1][2][3]. As a generalization of the FT, Fractional Fourier Transform (FRFT) has attracted wide attention in recent years.…”

Section: Introductionmentioning

confidence: 99%

2D-FRFT Based Frequency Shift-Invariant Digital Image Encryption

Gao

Guan

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In this paper, we study the property of frequency shift in two-dimensional Fractional Fourier Transform (2D-FRFT) domain. Based on the mathematical verification and computer simulations, it is demonstrated that the magnitude of reconstruction from phase information satisfies frequency shift-invariant in 2D-FRFT domain, improving the robustness of image encryption. Experiments are implemented to verify the effectiveness of this property against the frequency shift attack in 2D-FRFT domain, improving the robustness of image encryption.

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The Graphon Fourier Transform

Cited by 16 publications

References 26 publications

Graphon Filters: Graph Signal Processing in the Limit

Graphon Filters: Graph Signal Processing in the Limit

Graphon and Graph Neural Network Stability

2D-FRFT Based Frequency Shift-Invariant Digital Image Encryption

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