p-Laplacian Regularization of Signals on Directed Graphs

  • Zeina Abu Aisheh
  • Sébastien Bougleux
  • Olivier LézorayEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11241)


The graph Laplacian plays an important role in describing the structure of a graph signal from weights that measure the similarity between the vertices of the graph. In the literature, three definitions of the graph Laplacian have been considered for undirected graphs: the combinatorial, the normalized and the random-walk Laplacians. Moreover, a nonlinear extension of the Laplacian, called the p-Laplacian, has also been put forward for undirected graphs. In this paper, we propose several formulations for p-Laplacians on directed graphs directly inspired from the Laplacians on undirected graphs. Then, we consider the problem of p-Laplacian regularization of signals on directed graphs. Finally, we provide experimental results to illustrate the effect of the proposed p-laplacians on different types of graph signals.


Directed graphs p-laplacian Graph signal Regularization 


  1. 1.
    Bauer, F.: Normalized graph Laplacians for directed graphs. Linear Algebra Appl. 436(11), 4193–4222 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bougleux, S., Elmoataz, A., Melkemi, M.: Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. Int. J. Comput. Vis. 84(2), 220–236 (2009)CrossRefGoogle Scholar
  3. 3.
    Chung, F.R.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92, pp. 1–212 (1997)Google Scholar
  4. 4.
    Chung, F.: Laplacians and the cheeger inequality for directed graphs. Ann. Comb. 9(1), 1–19 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, 4th edn., vol. 173. Springer, Heidelberg (2012)Google Scholar
  6. 6.
    Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17(7), 1047–1060 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hein, M., Audibert, J., von Luxburg, U.: Graph Laplacians and their convergence on random neighborhood graphs. J. Mach. Learn. Res. 8, 1325–1368 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kheradmand, A., Milanfar, P.: A general framework for regularized, similarity-based image restoration. IEEE Trans. Image Process. 23(12), 5136–5151 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lézoray, O., Grady, L.: Image Processing and Analysis with Graphs: Theory and Practice. Digital Imaging and Computer Vision. CRC Press/Taylor and Francis, Boca Raton (2012)CrossRefGoogle Scholar
  10. 10.
    von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic, pp. 849–856 (2001)Google Scholar
  12. 12.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Automat. Contr. 49(9), 1520–1533 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pang, J., Cheung, G.: Graph laplacian regularization for image denoising: analysis in the continuous domain. IEEE Trans. Image Process. 26(4), 1770–1785 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shuman, D.I., Narang, S.K., Frossard, P., Ortega, A., Vandergheynst, P.: The emerging field of signal processing on graphs: extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process. Mag. 30(3), 83–98 (2013)CrossRefGoogle Scholar
  15. 15.
    Singh, R., Chakraborty, A., Manoj, B.S.: Graph Fourier transform based on directed Laplacian. In: 2016 International Conference on Signal Processing and Communications (SPCOM), pp. 1–5 (2016)Google Scholar
  16. 16.
    Tremblay, N., Gonçalves, P., Borgnat, P.: Design of graph filters and filterbanks. ArXiv e-prints (2017)Google Scholar
  17. 17.
    Zhou, D., Huang, J., Schölkopf, B.: Learning from labeled and unlabeled data on a directed graph. In: Proceedings of the Twenty-Second International Conference on Machine Learning (ICML 2005), Bonn, Germany, 7–11 August 2005, pp. 1036–1043 (2005)Google Scholar
  18. 18.
    Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) DAGM 2005. LNCS, vol. 3663, pp. 361–368. Springer, Heidelberg (2005). Scholar
  19. 19.
    Zhou, D., Schölkopf, B., Hofmann, T.: Semi-supervised learning on directed graphs. In: Advances in Neural Information Processing Systems 17 (Neural Information Processing Systems, NIPS 2004, 13–18 December 2004, Vancouver, British Columbia, Canada, pp. 1633–1640 (2004)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Zeina Abu Aisheh
    • 1
  • Sébastien Bougleux
    • 1
  • Olivier Lézoray
    • 1
    Email author
  1. 1.Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYCCaenFrance

Personalised recommendations