{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/rational-neural-networks-for-approximating","title":"Rational Neural Networks for Approximating Jump Discontinuities of Graph Convolution Operator","arxiv_id":"1808.10073","date":"2018-08-30","proceeding":null,"authors":["Zhiqian Chen","Feng Chen","Rongjie Lai","Xuchao Zhang","Chang-Tien Lu"],"abstract":"For node level graph encoding, a recent important state-of-art method is the\ngraph convolutional networks (GCN), which nicely integrate local vertex\nfeatures and graph topology in the spectral domain. However, current studies\nsuffer from several drawbacks: (1) graph CNNs relies on Chebyshev polynomial\napproximation which results in oscillatory approximation at jump\ndiscontinuities; (2) Increasing the order of Chebyshev polynomial can reduce\nthe oscillations issue, but also incurs unaffordable computational cost; (3)\nChebyshev polynomials require degree $\\Omega$(poly(1/$\\epsilon$)) to\napproximate a jump signal such as $|x|$, while rational function only needs\n$\\mathcal{O}$(poly log(1/$\\epsilon$))\\cite{liang2016deep,telgarsky2017neural}.\nHowever, it's non-trivial to apply rational approximation without increasing\ncomputational complexity due to the denominator. In this paper, the superiority\nof rational approximation is exploited for graph signal recovering. RatioanlNet\nis proposed to integrate rational function and neural networks. We show that\nrational function of eigenvalues can be rewritten as a function of graph\nLaplacian, which can avoid multiplication by the eigenvector matrix. Focusing\non the analysis of approximation on graph convolution operation, a graph signal\nregression task is formulated. Under graph signal regression task, its time\ncomplexity can be significantly reduced by graph Fourier transform. To overcome\nthe local minimum problem of neural networks model, a relaxed Remez algorithm\nis utilized to initialize the weight parameters. Convergence rate of\nRatioanlNet and polynomial based methods on jump signal is analyzed for a\ntheoretical guarantee. The extensive experimental results demonstrated that our\napproach could effectively characterize the jump discontinuities, outperforming\ncompeting methods by a substantial margin on both synthetic and real-world\ngraphs.","url_abs":"http://arxiv.org/abs/1808.10073v1","url_pdf":"http://arxiv.org/pdf/1808.10073v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"rational-neural-networks-for-approximating","repo_url":"https://github.com/aquastar/RationalGraphNet","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[{"method_slug":"convolution","method_name":"Convolution"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1808.10073","atlas_url":"https://app.syntology.ai/?focus=1808.10073","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}