Journal of Mathematical Imaging and Vision

, Volume 52, Issue 3, pp 414–435 | Cite as

Global Binary Optimization on Graphs for Classification of High-Dimensional Data

  • Ekaterina Merkurjev
  • Egil BaeEmail author
  • Andrea L. Bertozzi
  • Xue-Cheng Tai


This work develops a global minimization framework for segmentation of high-dimensional data into two classes. It combines recent convex optimization methods from imaging with recent graph- based variational models for data segmentation. Two convex splitting algorithms are proposed, where graph-based PDE techniques are used to solve some of the subproblems. It is shown that global minimizers can be guaranteed for semi-supervised segmentation with two regions. If constraints on the volume of the regions are incorporated, global minimizers cannot be guaranteed, but can often be obtained in practice and otherwise be closely approximated. Experiments on benchmark data sets show that our models produce segmentation results that are comparable with or outperform the state-of-the-art algorithms. In particular, we perform a thorough comparison to recent MBO (Merriman–Bence–Osher, AMS-Selected Lectures in Mathematics Series: Computational Crystal Growers Workshop, 1992) and phase field methods, and show the advantage of the algorithms proposed in this paper.


Classification Graphs Combinatorial optimization  Convex optimization 



E. Bae is supported by the Norwegian Research Council eVita Project 214889. E. Merkurjev is supported by the National Science Foundation Graduate Fellowship (NSF). X.-C. Tai is supported by the Christian Michelsen Research (CMR), Bergen. This work was supported by AFOSR MURI Grant FA9550-10-1-0569, NSF Grant DMS-111897, ONR Grants N000141210040 and N000141210838, and the W. M. Keck Foundation.


  1. 1.
    Anderson, C.: A Rayleigh–Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices. J. Comput. Phys. 229, 7477–7487 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bache, K., Lichman, M.: UCI machine learning repository (2013).
  3. 3.
    Belkin, M., Niyogi, P., Sindhwani, V.: Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. J. Mach. Learn. Res. 7, 2399–2434 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bertozzi, A., Flenner, A.: Diffuse interface models on graphs for classification of high dimensional data. Multiscale Model. Simul. 10(3), 1090–1118 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Trans. Pattern Anal. Mach. Intell. 26, 359–374 (2001)Google Scholar
  6. 6.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001)CrossRefGoogle Scholar
  7. 7.
    Bresson, X., Chan, T.F.: Non-local unsupervised variational image segmentation models. Tech. Rep., UCLA, cam-report 08–67 (2008)Google Scholar
  8. 8.
    Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J., Osher, S.: Fast global minimization of the active contour/snake model. J. Math Imaging Vis. 28(2), 151–167 (2007)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.: Multiclass total variation clustering. Advances in Neural Information Processing Systems, pp. 1421–1429. ACM, New York (2013)Google Scholar
  10. 10.
    Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.H.: Convergence and energy landscape for Cheeger cut clustering. Adv. Neural Inf. Process. Syst. 25, 1394–1402 (2012)Google Scholar
  11. 11.
    Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.H.: An adaptive total variation algorithm for computing the balanced cut of a graph. arXiv preprint arXiv:1302.2717 (2013)
  12. 12.
    Bresson, X., Tai, X.C., Chan, T.F., Szlam, A.: Multi-class transductive learning based on \(l^1\) relaxations of cheeger cut and mumford-shah-potts model. J. Math. Imaging Vis. 49(1), 191–201 (2014)Google Scholar
  13. 13.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1), 89–97 (2004)MathSciNetGoogle Scholar
  15. 15.
    Chan, T., Vese, L.: Active contours without edges. IEEE Image Proc. 10, 266–277 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Chan, T.F., Esedo\(\bar{{\rm g}}\)lu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)Google Scholar
  17. 17.
    Chapelle, O., Schölkopf, B., Zien, A.: Semi-Supervised Learning, vol. 2. MIT Press, Cambridge (2006)CrossRefGoogle Scholar
  18. 18.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Princeton University Press, Princeton (1970)Google Scholar
  19. 19.
    Chung, F.: Spectral Graph Theory, vol. 92. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  20. 20.
    Cireşan, D., Meier, U., Masci, J., Gambardella, L., Schmidhuber, J.: Flexible, high performance convolutional neural networks for image classification. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence, pp. 1237–1242 (2011)Google Scholar
  21. 21.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watershed: a unifying graph-based optimization framework. IEEE Trans. Pattern Anal. Mach. Intell. 33(7), 1384–1399 (2011)CrossRefGoogle Scholar
  22. 22.
    Couprie, C., Grady, L., Talbot, H., Najman, L.: Combinatorial continuous maximum flow. SIAM J. Imaging Sci. 4(3), 905–930 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Decoste, D., Schölkopf, B.: Training invariant support vector machines. Mach. Learn. 46(1), 161–190 (2002)CrossRefzbMATHGoogle Scholar
  24. 24.
    Ekeland, I., Téman, R.: Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  25. 25.
    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)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Esser, J.E.: Primal dual algorithms for convex models and applications to image restoration, registration and nonlocal inpainting (Ph.D. thesis, UCLA CAM-report 10–31) (2010)Google Scholar
  27. 27.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  28. 28.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split bregman method: segmentation and surface reconstruction. J. Sci. Comput. 45(1), 271–293 (2010)MathSciNetGoogle Scholar
  31. 31.
    Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Grady, L.: Multilabel random walker image segmentation using prior models. In: Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 763–770 (2005)Google Scholar
  33. 33.
    Grady, L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1768–1783 (2006)CrossRefGoogle Scholar
  34. 34.
    Grady, L., Polimeni, J.R.: Discrete Calculus: Applied Analysis on Graphs for Computational Science. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  35. 35.
    Grady, L., Schiwietz, T., Aharon, S., Westermann, R.: Random walks for interactive alpha-matting. In: Proceedings of VIIP, pp. 423–429 (2005)Google Scholar
  36. 36.
    Hein, M., Audibert, J., Von Luxburg, U.: From graphs to manifolds—weak and strong pointwise consistency of graph laplacians. In: Proceedings of the 18th Conference on Learning Theory (COLT, pp. 470–485). Springer, Berlin (2005)Google Scholar
  37. 37.
    Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. Adv. Neural Inf. Process. Syst. 23, 847–855 (2010)Google Scholar
  38. 38.
    Hu, H., Laurent, T., Porter, M.A., Bertozzi, A.L.: A method based on total variation for network modularity optimization using the MBO scheme. SIAM J. Appl. Math. 73(6), 2224–2246 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Kégl, B., Busa-Fekete, R.: Boosting products of base classifiers. In: Proceedings of the 26th Annual International Conference on Machine Learning, pp. 497–504 (2009)Google Scholar
  40. 40.
    Klodt, M., Cremers, D.: A convex framework for image segmentation with moment constraints. In: IEEE International Conference on Computer Vision (ICCV), pp. 2236–2243 (2011)Google Scholar
  41. 41.
    Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 26, 65–81 (2004)CrossRefGoogle Scholar
  42. 42.
    LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)CrossRefGoogle Scholar
  43. 43.
    LeCun, Y., Cortes, C.: The MNIST database of handwritten digits.
  44. 44.
    Levin, A., Rav-Acha, A., Lischinski, D.: Spectral matting. IEEE Trans. Pattern Anal. Mach. Intell. 30(10), 1699–1712 (2008)CrossRefGoogle Scholar
  45. 45.
    Lezoray, O., Elmoataz, A., Ta, V.T.: Nonlocal pdes on graphs for active contours models with applications to image segmentation and data clustering. In: Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on, pp. 873–876. IEEE (2012)Google Scholar
  46. 46.
    Merkurjev, E., Kostic, T., Bertozzi, A.: An MBO scheme on graphs for classification and image processing. SIAM J. Imaging Sci. 6(4), 1903–1930 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Merriman, B., Bence, J.K., Osher, S.: Diffusion generated motion by mean curvature. AMS Selected Lectures in Mathematics Series: Computational Crystal Growers Workshop 8966, 73–83 (1992)Google Scholar
  48. 48.
    Mohar, B.: The Laplacian spectrum of graphs. Graph Theory Comb. Appl. 2, 871–898 (1991)MathSciNetGoogle Scholar
  49. 49.
    Mollenhoff, T., Nieuwenhuis, C., Toppe, E., Cremers, D.: Efficient convex optimization for minimal partition problems with volume constraints. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 94–107 (2013)Google Scholar
  50. 50.
    Perona, P., Zelnik-Manor, L.: Self-tuning spectral clustering. Adv. Neural Inf. Process. Syst. 17, 1601–1608 (2004)Google Scholar
  51. 51.
    Setzer, S.: Operator splittings, bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  53. 53.
    Szlam, A., Bresson, X.: Total variation and cheeger cuts. In: J.Fürnkranz, T. Joachims (eds.) Proceedings of the 27th International Conference on Machine Learning (ICML-10), pp. 1039–1046. Omnipress, Haifa, Israel (2010). URL
  54. 54.
    Szlam, A., Bresson, X.: A total variation-based graph clustering algorithm for Cheeger ratio cuts. In: Proceedings of the 27th International Conference on Machine Learning, pp. 1039–1046 (2010)Google Scholar
  55. 55.
    Szlam, A.D., Maggioni, M., Coifman, R.R.: Regularization on graphs with function-adapted diffusion processes. J. Mach. Learn. Res. 9, 1711–1739 (2008)zbMATHMathSciNetGoogle Scholar
  56. 56.
    Tai, X.C., Christiansen, O., Lin, P., Skjælaaen, I.: Image segmentation using some piecewise constant level set methods with mbo type of projection. Int. J. Comput. Vis. 73(1), 61–76 (2007)CrossRefGoogle Scholar
  57. 57.
    van Gennip, Y., Bertozzi, A.: Gamma-convergence of graph Ginzburg–Landau functionals. Advanced in Differential Equations 17(11–12), 1115–1180 (2012)zbMATHGoogle Scholar
  58. 58.
    van Gennip, Y., Guillen, N., Osting, B., Bertozzi, A.L.: Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan J. Math. 82(1), 3–65 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Wang, J., Jebara, T., Chang, S.: Graph transduction via alternating minimization. In: Proceedings of the 25th International Conference on Machine Learning, pp. 1144–1151 (2008)Google Scholar
  60. 60.
    Wu, C., Tai, X.C.: Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Yuan, J., Bae, E., Tai, X.C.: A study on continuous max-flow and min-cut approaches. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition, pp. 2217–2224 (2010)Google Scholar
  62. 62.
    Yuan, J., Bae, E., Tai, X.C., Boykov, Y.: A spatially continuous max-flow and min-cut framework for binary labeling problems. Numer. Math. 126(3), 559–587 (2013)CrossRefMathSciNetGoogle Scholar
  63. 63.
    Zhou, D., Bousquet, O., Lal, T.N., Weston, J., Schölkopf, B.: Learning with local and global consistency. Adv. Neural Inf. Process. Syst. 16, 321–328 (2004)Google Scholar
  64. 64.
    Zhou, D., Schölkopf, B.: A regularization framework for learning from graph data. In: Workshop on Statistical Relational Learning. International Conference on Machine Learning (2004)Google Scholar
  65. 65.
    Zhu, X.: Semi-supervised learning literature survey. Computer Sciences Technical Report 1530. University of Wisconsin, Madison (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Ekaterina Merkurjev
    • 1
  • Egil Bae
    • 1
    Email author
  • Andrea L. Bertozzi
    • 1
  • Xue-Cheng Tai
    • 2
  1. 1.University of CaliforniaLos AngelesUSA
  2. 2.University of BergenBergenNorway

Personalised recommendations