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Wavelets on Graphs via Spectral Graph Theory
TLDR
In this paper, the spectral graph wavelet operator is defined based on spectral decomposition of the discrete graph Laplacian, and a wavelet generating kernel and a scale parameter are used to localize this operator to an indicator function.Abstract:
We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $\L$. Given a wavelet generating kernel $g$ and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(t\L)$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $\L$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.read more
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Semi-Supervised Classification with Graph Convolutional Networks
Thomas Kipf,Max Welling +1 more
TL;DR: A scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs which outperforms related methods by a significant margin.
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Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering
TL;DR: In this article, a spectral graph theory formulation of convolutional neural networks (CNNs) was proposed to learn local, stationary, and compositional features on graphs, and the proposed technique offers the same linear computational complexity and constant learning complexity as classical CNNs while being universal to any graph structure.
Journal ArticleDOI
The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains
TL;DR: The field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process high-dimensional data on graphs as discussed by the authors, which are the analogs to the classical frequency domain and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs.
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Graph Neural Networks: A Review of Methods and Applications
Jie Zhou,Ganqu Cui,Shengding Hu,Zhengyan Zhang,Cheng Yang,Zhiyuan Liu,Lifeng Wang,Changcheng Li,Maosong Sun +8 more
TL;DR: A detailed review over existing graph neural network models is provided, systematically categorize the applications, and four open problems for future research are proposed.
Proceedings ArticleDOI
Spatio-temporal graph convolutional networks: a deep learning framework for traffic forecasting
Bing Yu,Haoteng Yin,Zhanxing Zhu +2 more
TL;DR: Wang et al. as mentioned in this paper proposed a novel deep learning framework, Spatio-Temporal Graph Convolutional Networks (STGCN), to tackle the time series prediction problem in traffic domain.
References
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Shiftable multiscale transforms
TL;DR: Two examples of jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored and the usefulness of these image representations for scale-space analysis, stereo disparity measurement, and image enhancement is demonstrated.
Journal ArticleDOI
Continuous and discrete wavelet transforms
Christopher Heil,David F. Walnut +1 more
TL;DR: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states, focusing on Weyl–Heisenberg coherent states and affine coherent states.
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A global, self‐consistent, hierarchical, high‐resolution shoreline database
Paul Wessel,Walter H. F. Smith +1 more
TL;DR: A high-resolution shoreline data set amalgamated from two databases in the public domain, which has undergone extensive processing and is free of internal inconsistencies such as erratic points and crossing segments is presented.
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Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency
L. Sendur,Ivan Selesnick +1 more
TL;DR: This work proposes new non-Gaussian bivariate distributions, and corresponding nonlinear threshold functions (shrinkage functions) are derived from the models using Bayesian estimation theory, but the new shrinkage functions do not assume the independence of wavelet coefficients.
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Wavelet analysis and synthesis of fractional Brownian motion
TL;DR: A detailed second-order analysis is carried out for wavelet coefficients of FBM, revealing a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of F BM can be estimated.