A path following algorithm for the graph matching problem

Mikhail Zaslavskiy (1, 2) , Francis Bach (1, 3) , Jean-Philippe Vert (2)

1 CMM - Centre de Morphologie Mathématique
2 CBIO - Centre de Bioinformatique
3 WILLOW - Models of visual object recognition and scene understanding

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore to perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four datasets: simulated graphs, QAPLib, retina vessel images and handwritten chinese characters. In all cases, the results are competitive with the state-of-the-art.

Keywords: graph matching; graph algorithms; convex programming; gradient methods; machine learning; classification; image processing

Domains: Other Statistics [stat.ML]; Optimization and Control [math.OC]; Computer Vision and Pattern Recognition [cs.CV]

Mikhail Zaslavskiy, Francis Bach, Jean-Philippe Vert. A path following algorithm for the graph matching problem. 2008. ⟨hal-00232851⟩