A Fast MBO Scheme for Multiclass Data Classification

  • Matt JacobsEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


We describe a new variant of the MBO scheme for solving the semi-supervised data classification problem on a weighted graph. The scheme is based on the minimization of the graph heat content energy. The resulting algorithms guarantee dissipation of the graph heat content energy for an extremely wide class of weight matrices. As a result, our method is both flexible and unconditionally stable. Experimental results on benchmark machine learning datasets show that our approach matches or exceeds the performance of current state-of-the-art variational methods while being considerably faster.


Weight Matrix Weighted Graph Geodesic Distance Affinity Matrix Local Scaling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author is grateful to Selim Esedoḡlu for helpful comments and suggestions. The author was supported by NSF DMS-1317730.


  1. 1.
    Alberti, G., Bellettini, G.: A non-local anisotropic model for phase transitions: asymptotic behavior of rescaled energies. Eur. J. Appl. Math. 9, 261–284 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bresson, X., Chan, T., Tai, X., Szlam, A.: Multi-class trans- ductive learning based on l1 relaxations of cheeger cut and mumford-shah-potts model. J. Math. Imaging Vis. 49(1), 191–201 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis, pp. 195–199 (1970)Google Scholar
  4. 4.
    Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.W.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. PNAS 102, 7426–7431 (2005)CrossRefGoogle Scholar
  5. 5.
    Elsey, M., Esedo\(\bar{\text{g}}\)lu, S.: Threshold dynamics for anisotropic surface energies. Technical report, UM (2016). Under reviewGoogle Scholar
  6. 6.
    Esedo\(\bar{\text{ g }}\)lu, S., Otto, F.: Threshold dynamics for networks with arbitrary surface tensions. Commun. Pure Appl. Math. 68(5), 808–864 (2015)Google Scholar
  7. 7.
    Esedo\(\bar{\text{ g }}\)lu, S., Tsai, Y.-H.: Threshold dynamics for the piecewise constant Mumford-Shah functional. J. Comput. Phys. 211(1), 367–384 (2006)Google Scholar
  8. 8.
    Esedo\(\bar{\text{ g }}\)lu, S., Jacobs, M.: Convolution kernels, and stability of threshold dynamics methods. Technical report, University of Michigan (2016)Google Scholar
  9. 9.
    Garcia-Cardona, C., Merkurjev, E., Bertozzi, A.L., Flenner, A., Percus, A.G.: Multiclass data segmentation using diffuse interface methods on graphs. IEEE Trans. Pattern Anal. Mach. Intell. 36(8), 1600–1613 (2014)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hein, M., Setzer, S.: Beyond spectral clustering - tight relaxations of balanced graph cuts. In: Advances in Neural Information Processing Systems 24 (NIPS) (2011)Google Scholar
  11. 11.
    Kaynak, C.: Methods of combining multiple classifiers and their applications to handwritten digit recognition. Master’s thesis, Institute of Graduate Studies in Science and Engineering, Bogazici University (1995)Google Scholar
  12. 12.
    Malik, J., Shi, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  13. 13.
    Merkurjev, E., Bertozzi, A., Chung, F.: A semi-supervised heat kernel pagerank MBO algorithm for data classification (2016, submitted)Google Scholar
  14. 14.
    Merriman, B., Bence, J.K., Osher, S.J.: Diffusion generated motion by mean curvature. In: Taylor, J. (ed.) Proceedings of the Computational Crystal Growers Workshop, pp. 73–83. AMS (1992)Google Scholar
  15. 15.
    Miranda, M., Pallara, D., Paronetto, F., Preunkert, M.: Short-time heat flow and functions of bounded variation in \({\mathbb{R}}^N\). Ann. Fac. Sci. Toulouse Math. 16(1), 125–145 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nene, S.A., Nayar, S.K., Murase, H.: Columbia object image library (coil-100). Technical report, Columbia University (1996)Google Scholar
  18. 18.
    Zien, A., Chapelle, O., Scholkopf, B.: Semi-Supervised Learning. The MIT Press, Cambridge (2006)Google Scholar
  19. 19.
    Scherzer, O. (ed.): Handbook of Mathematical Methods in Imaging. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  20. 20.
    Yin, K., Tai, X.-Y., Osher, S.J.: An effective region force for some variational models for learning and clustering. Technical report, UCLA (2016)Google Scholar
  21. 21.
    Yu, S.X., Shi, J.: Multiclass spectral clustering. In: Ninth IEEE International Coference on Computer Vision, vol. 1, pp. 313–319, October 2003Google Scholar
  22. 22.
    Zelnik-Manor, L., Perona, P.: Self-tuning spectral clustering. In: Advances in Neural Information Processing Systems (2004)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA

Personalised recommendations