Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Kschischang, F., Frey, B., Loeliger, H.A.: Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory 47, 498–519 (2001)
Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1, 1–305 (2008)
Schlesinger, M.: Syntactic analysis of two-dimensional visual signals in the presence of noise. Kibernetika, 113–130 (1976)
Werner, T.: A linear programming approach to max-sum problem: A review. IEEE PAMI 29 (2007)
Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. IEEE PAMI 28, 1568–1583 (2006)
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)
Komodakis, N., Paragios, N., Tziritas, G.: MRF optimization via dual decomposition: Message-passing revisited. In: ICCV (2007)
Schlesinger, M., Giginyak, V.: Solution to structural recognition (MAX,+)-problems by their equivalent transformations. In 2 parts. Control Systems and Computers (2007)
Savchynskyy, B., Kappes, J., Schmidt, S., Schnörr, C.: A study of Nesterov’s scheme for Lagrangian decomposition and MAP labeling. In: CVPR 2011 (2011)
Korte, B., Vygen, J.: Combinatorial Optimization, 4th edn. Springer, Heidelberg (2008)
Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. Ser. A, 127–152 (2004)
Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. The John Hopkins Univ. Press, Baltimore (1996)
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)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schmidt, S., Savchynskyy, B., Kappes, J.H., Schnörr, C. (2011). Evaluation of a First-Order Primal-Dual Algorithm for MRF Energy Minimization. In: Boykov, Y., Kahl, F., Lempitsky, V., Schmidt, F.R. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2011. Lecture Notes in Computer Science, vol 6819. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23094-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-23094-3_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23093-6
Online ISBN: 978-3-642-23094-3
eBook Packages: Computer ScienceComputer Science (R0)