Issue No. 12 - December (2009 vol. 31)

ISSN: 0162-8828

pp: 2227-2242

Mikhail Zaslavskiy , Mines ParisTech and Institute Curie-INSERM, France

Jean-Philippe Vert , Mines ParisTech and Institute Curie-INSERM, France

Francis Bach , INRIA-d'Informatique de l'Ecole Normale Supérieure, Paris

ABSTRACT

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, perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four data sets: simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. In all cases, the results are competitive with the state of the art.

INDEX TERMS

Graph algorithms, graph matching, convex programming, gradient methods, machine learning, classification, image processing.

CITATION

Mikhail Zaslavskiy, Jean-Philippe Vert, Francis Bach, "A Path Following Algorithm for the Graph Matching Problem",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol. 31, no. , pp. 2227-2242, December 2009, doi:10.1109/TPAMI.2008.245