Evaluation of a First-Order Primal-Dual Algorithm for MRF Energy Minimization

  • Stefan Schmidt
  • Bogdan Savchynskyy
  • Jörg Hendrik Kappes
  • Christoph Schnörr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6819)


We investigate the First-Order Primal-Dual (FPD) algorithm of Chambolle and Pock [1] in connection with MAP inference for general discrete graphical models. We provide a tight analytical upper bound of the stepsize parameter as a function of the underlying graphical structure (number of states, graph connectivity) and thus insight into the dependency of the convergence rate on the problem structure. Furthermore, we provide a method to compute efficiently primal and dual feasible solutions as part of the FPD iteration, which allows to obtain a sound termination criterion based on the primal-dual gap. An experimental comparison with Nesterov’s first-order method in connection with dual decomposition shows superiority of the latter one in optimizing the dual problem. However due to the direct optimization of the primal bound, for small-sized (e.g. 20x20 grid graphs) problems with a large number of states, FPD iterations lead to faster improvement of the primal bound and a resulting faster overall convergence.


graphical model MAP inference LP relaxation image labeling sparse convex programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging and Vision, 1–26 (2010)Google Scholar
  2. 2.
    Kschischang, F., Frey, B., Loeliger, H.A.: Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory 47, 498–519 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1, 1–305 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Schlesinger, M.: Syntactic analysis of two-dimensional visual signals in the presence of noise. Kibernetika, 113–130 (1976)Google Scholar
  5. 5.
    Werner, T.: A linear programming approach to max-sum problem: A review. IEEE PAMI 29 (2007)Google Scholar
  6. 6.
    Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. IEEE PAMI 28, 1568–1583 (2006)CrossRefGoogle Scholar
  7. 7.
    Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., Tappen, M., Rother, C.: A comparative study of energy minimization methods for Markov random fields with smoothness-based priors. IEEE PAMI 30, 1068–1080 (2008)CrossRefGoogle Scholar
  8. 8.
    Komodakis, N., Paragios, N., Tziritas, G.: MRF optimization via dual decomposition: Message-passing revisited. In: ICCV (2007)Google Scholar
  9. 9.
    Schlesinger, M., Giginyak, V.: Solution to structural recognition (MAX,+)-problems by their equivalent transformations. In 2 parts. Control Systems and Computers (2007)Google Scholar
  10. 10.
    Savchynskyy, B., Kappes, J., Schmidt, S., Schnörr, C.: A study of Nesterov’s scheme for Lagrangian decomposition and MAP labeling. In: CVPR 2011 (2011)Google Scholar
  11. 11.
    Korte, B., Vygen, J.: Combinatorial Optimization, 4th edn. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  12. 12.
    Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. Ser. A, 127–152 (2004)Google Scholar
  13. 13.
    Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. The John Hopkins Univ. Press, Baltimore (1996)zbMATHGoogle Scholar
  14. 14.
    Michelot, C.: A finite algorithm for finding the projection of a point onto the canonical simplex of Rn. J. Optim. Theory Appl. 50, 195–200 (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Stefan Schmidt
    • 1
  • Bogdan Savchynskyy
    • 1
  • Jörg Hendrik Kappes
    • 1
  • Christoph Schnörr
    • 1
  1. 1.IWR / HCIHeidelberg UniversityHeidelbergGermany

Personalised recommendations