On The Stability of Polynomial Spectral Graph Filters

Kenlay, Henry; Thanou, Dorina; Dong, Xiaowen

doi:10.1109/icassp40776.2020.9054072

Cited by 19 publications

(28 citation statements)

References 33 publications

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“…We then look at the filter distance for fractional order polynomial GF of order 0.2, 0.5 and 0.8. As shown in Figure 7, the filter distance can be seen to scale linearly with the Laplacian distance consistent with Theorem 2 in [46] which states that polynomial filters are linearly stable.…”

Section: Stability Analysissupporting

confidence: 78%

“…When utilizing spectral GFs for learning representations, a necessary condition for transferability in certain tasks is stability. Stability is defined to be the property such that if we add a small perturbation to the input graph, the output of the filter is also perturbed by a small amount [46].…”

Section: Stability Analysismentioning

confidence: 99%

“…The authors of [46] prove that polynomial GFs are stable with respect to the change in the normalized graph Laplacian matrix. They defined filter distance as…”

Section: Stability Analysismentioning

confidence: 99%

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Fractional Order Graph Filters: Design and Implementation

Qiu

Feng

2021

Electronics

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Existing graph filters, polynomial or rational, are mainly of integer order forms. However, there are some frequency responses which are not easily achieved by integer order approximation. It will substantially increase the flexibility of the filters if we relax the integer order to fractional ones. Motivated by fractional order models, we introduce the fractional order graph filters (FOGF), and propose to design the filter coefficients by genetic algorithm. In order to implement distributed computation on a graph, an FOGF can be approximated by the continued fraction expansion and transformed to an infinite impulse response graph filter.

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Section: Stability Analysissupporting

confidence: 78%

Section: Stability Analysismentioning

confidence: 99%

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Fractional Order Graph Filters: Design and Implementation

Qiu

Feng

2021

Electronics

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“…Our main goal is to extend bounds found in the existing literature so that they have an interpretation in terms of the structural properties of the existing and perturbed graphs. To do this, we focus on polynomial graph filters, and build on our previous work [13], by providing a new bound that is tighter and generalises to the family of normalised augmented adjacency matrices. We then bound the change in normalised augmented adjacency matrix by considering the largest change around each node where the change admits a structural interpretation.…”

Section: Introductionmentioning

confidence: 99%

On The Stability of Graph Convolutional Neural Networks Under Edge Rewiring

Kenlay

Thano

Dong

2021

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

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Graph neural networks are experiencing a surge of popularity within the machine learning community due to their ability to adapt to non-Euclidean domains and instil inductive biases. Despite this, their stability, i.e., their robustness to small perturbations in the input, is not yet well understood. Although there exists some results showing the stability of graph neural networks, most take the form of an upper bound on the magnitude of change due to a perturbation in the graph topology. However, the change in the graph topology captured in existing bounds tend not to be expressed in terms of structural properties, limiting our understanding of the model robustness properties. In this work, we develop an interpretable upper bound elucidating that graph neural networks are stable to rewiring between high degree nodes. This bound and further research in bounds of similar type provide further understanding of the stability properties of graph neural networks.

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“…Therefore, in practice, we expect that a node can only impose direct influence on an adjacent node through the shift operator. For practical design purposes, it is advantageous to be able to decompose filters in a form of a polynomial of such a shift matrix [15]. An example is shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

Undirected Graphs: Is the Shift-Enabled Condition Trivial or Necessary?

Chen

Cheng

et al. 2021

IEEE Access

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With the growing application of undirected graphs for signal/image processing on graphs and distributed machine learning, we demonstrate that the shift-enabled condition is as necessary for undirected graphs as it is for directed graphs. It has recently been shown that, contrary to the widespread belief that a shift-enabled condition (necessary for any shift-invariant filter to be representable by a graph shift matrix) can be ignored because any non-shift-enabled matrix can be converted to a shift-enabled matrix, such a conversion in general may not hold for a directed graph with non-symmetric shift matrix. This paper extends this prior work, focusing on undirected graphs where the shift matrix is generally symmetric. We show that while, in this case, the shift matrix can be converted to satisfy the original shift-enabled condition, the converted matrix is not associated with the original graph, that is, it does not capture anymore the structure of the graph signal. We show via examples, that a non-shift-enabled matrix cannot be converted to a shiftenabled one and still maintain the topological structure of the underlying graph, which is necessary to facilitate localized signal processing.

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On The Stability of Polynomial Spectral Graph Filters

Cited by 19 publications

References 33 publications

Fractional Order Graph Filters: Design and Implementation

Fractional Order Graph Filters: Design and Implementation

On The Stability of Graph Convolutional Neural Networks Under Edge Rewiring

Undirected Graphs: Is the Shift-Enabled Condition Trivial or Necessary?

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