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Graph Fourier transform based on singular value decomposition of the directed Laplacian

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Abstract

The Graph Fourier transform (GFT) is a fundamental tool in graph signal processing. In this paper, based on singular value decomposition of the Laplacian, we introduce a novel definition of GFT on directed graphs, and use the singular values of the Laplacian to carry the notion of graph frequencies. We show that the proposed GFT has its frequencies and frequency components evaluated by solving some constrained minimization problems with low computational cost, and it could represent graph signals with different modes of variation efficiently. Moreover, the proposed GFT is consistent with the conventional GFT in the undirected graph setting, and on directed circulant graphs, it is the classical discrete Fourier transform, up to some rotation, permutation and phase adjustment.

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Acknowledgements

The authors would like to thank the anonymous reviewers for constructive suggestions and insightful comments for the improvement.

Author information

Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, China

    Yang Chen

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, Guangdong, China

    Cheng Cheng

  3. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Qiyu Sun

Authors
  1. Yang Chen
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  2. Cheng Cheng
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  3. Qiyu Sun
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Correspondence to Qiyu Sun.

Additional information

This work is partially supported by the National Science Foundation (DMS-1816313), National Nature Science Foundation of China (11901192, 12171490), Guangdong Province Nature Science Foundation (2022A1515011060), and Guangzhou Science and Technology Foundation Committee (202201011126).

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Chen, Y., Cheng, C. & Sun, Q. Graph Fourier transform based on singular value decomposition of the directed Laplacian. Sampl. Theory Signal Process. Data Anal. 21, 24 (2023). https://doi.org/10.1007/s43670-023-00062-w

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