Abstract
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.
This work received funding from the Agence Nationale de la Recherche (ANR-14-CE27-0001 GRAPHSIP), and from the European Union FEDER/FSE 2014/2020 (GRAPHSIP project).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bauer, F.: Normalized graph Laplacians for directed graphs. Linear Algebra Appl. 436(11), 4193–4222 (2012)
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)
Chung, F.R.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92, pp. 1–212 (1997)
Chung, F.: Laplacians and the cheeger inequality for directed graphs. Ann. Comb. 9(1), 1–19 (2005)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, 4th edn., vol. 173. Springer, Heidelberg (2012)
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)
Hein, M., Audibert, J., von Luxburg, U.: Graph Laplacians and their convergence on random neighborhood graphs. J. Mach. Learn. Res. 8, 1325–1368 (2007)
Kheradmand, A., Milanfar, P.: A general framework for regularized, similarity-based image restoration. IEEE Trans. Image Process. 23(12), 5136–5151 (2014)
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)
von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)
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)
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)
Pang, J., Cheung, G.: Graph laplacian regularization for image denoising: analysis in the continuous domain. IEEE Trans. Image Process. 26(4), 1770–1785 (2017)
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)
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)
Tremblay, N., Gonçalves, P., Borgnat, P.: Design of graph filters and filterbanks. ArXiv e-prints (2017)
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)
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). https://doi.org/10.1007/11550518_45
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Abu Aisheh, Z., Bougleux, S., Lézoray, O. (2018). p-Laplacian Regularization of Signals on Directed Graphs. In: Bebis, G., et al. Advances in Visual Computing. ISVC 2018. Lecture Notes in Computer Science(), vol 11241. Springer, Cham. https://doi.org/10.1007/978-3-030-03801-4_57
Download citation
DOI: https://doi.org/10.1007/978-3-030-03801-4_57
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-03800-7
Online ISBN: 978-3-030-03801-4
eBook Packages: Computer ScienceComputer Science (R0)