Filter Design for Autoregressive Moving Average Graph Filters

We investigate a scalable M -channel critically sampled filter bank for graph signals, where each of the M filters is supported on a different subband of the graph Laplacian spectrum. For analysis, the graph signal is filtered on each subband and downsampled on a corresponding set of vertices. However, the classical synthesis filters are replaced with interpolation operators. For small graphs, we use a full eigendecomposition of the graph Laplacian to partition the graph vertices such that the m th set comprises a uniqueness set for signals supported on the m th subband. The resulting transform is critically sampled, the dictionary atoms are orthogonal to those supported on different bands, and graph signals are perfectly reconstructable from their analysis coefficients. We also investigate fast versions of the proposed transform that scale efficiently for large, sparse graphs. Issues that arise in this context include designing the filter bank to be more amenable to polynomial approximation, estimating the number of samples required for each band, performing non-uniform random sampling for the filtered signals on each band, and using efficient reconstruction methods. We empirically explore the joint vertex-frequency localization of the dictionary atoms, the sparsity of the analysis coefficients for different classes of signals, the reconstruction error resulting from the numerical approximations, and the ability of the proposed transform to compress piecewise-smooth graph signals. The proposed filter bank also yields a fast, approximate graph Fourier transform with a coarse resolution in the spectral domain.

Section: Conclusion and Extensionsmentioning

confidence: 99%

Scalable $M$-Channel Critically Sampled Filter Banks for Graph Signals

Jin

Shuman

2019

“…This implies that application of the filter Gf can be performed without explicit expensive eigendecomposition of the Laplacian operator. Cayley filters are special cases of filters based on general rational functions of the Laplacian, namely ARMA filters [18] [19]. For a general rational functions of the Laplacian, calculating the denominator requires a matrix inversion.…”

Section: Cayley Filtersmentioning

confidence: 99%

CayleyNets: Graph Convolutional Neural Networks With Complex Rational Spectral Filters

Levie

Monti

Bresson

et al. 2019

521

319

The rise of graph-structured data such as social networks, regulatory networks, citation graphs, and functional brain networks, in combination with resounding success of deep learning in various applications, has brought the interest in generalizing deep learning models to non-Euclidean domains. In this paper, we introduce a new spectral domain convolutional architecture for deep learning on graphs. The core ingredient of our model is a new class of parametric rational complex functions (Cayley polynomials) allowing to efficiently compute spectral filters on graphs that specialize on frequency bands of interest. Our model generates rich spectral filters that are localized in space, scales linearly with the size of the input data for sparsely-connected graphs, and can handle different constructions of Laplacian operators. Extensive experimental results show the superior performance of our approach, in comparison to other spectral domain convolutional architectures, on spectral image classification, community detection, vertex classification and matrix completion tasks. * The first two authors have contributed equally and are listed alphabetically. Ron Levie is with the Institute

“…Remark 4. The approximation accuracy of the EV ARMA 1 filters can be further improved by following the Shank's method [39] used in [10,38], or the iterative least-squares approach proposed in [40]. These methods have shown to improve the approximation accuracy of Prony's design by not only taking the modified fitting error into account but also the true one.…”

Section: Filter Designmentioning

confidence: 99%

Advances in Distributed Graph Filtering

Coutiño

Isufi

Leus

2019

Self Cite

112

Graph filters are one of the core tools in graph signal processing. A central aspect of them is their direct distributed implementation. However, the filtering performance is often traded with distributed communication and computational savings. To improve this tradeoff, this work generalizes state-of-theart distributed graph filters to filters where every node weights the signal of its neighbors with different values while keeping the aggregation operation linear. This new implementation, labeled as edge-variant graph filter, yields a significant reduction in terms of communication rounds while preserving the approximation accuracy. In addition, we characterize the subset of shift-invariant graph filters that can be described with edge-variant recursions. By using a low-dimensional parametrization the proposed graph filters provide insights in approximating linear operators through the succession and composition of local operators, i.e., fixed support matrices, which span applications beyond the field of graph signal processing. A set of numerical results shows the benefits of the edge-variant filters over current methods and illustrates their potential to a wider range of applications than graph filtering.