A Path Following Algorithm for Graph Matching

We propose a convex-concave programming approach for the labelled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the 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 aslo a complex combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of the convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. The algorithm is compared with some of the best performing graph matching methods on three datasets: simulated graphs, QAPLib and handwritten chinese characters.

[1]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[2]  Jitendra Malik,et al.  Shape matching and object recognition using shape contexts , 2010, 2010 3rd International Conference on Computer Science and Information Technology.

[3]  Terry Caelli,et al.  An eigenspace projection clustering method for inexact graph matching , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  M. Zaslavskiy,et al.  A Path Following Algorithm for the Graph Matching Problem , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Alexander Filatov,et al.  Graph-based handwritten digit string recognition , 1995, Proceedings of 3rd International Conference on Document Analysis and Recognition.

[6]  Gene H. Golub,et al.  Matrix computations , 1983 .

[7]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[8]  Christoph Schnörr,et al.  Probabilistic Subgraph Matching Based on Convex Relaxation , 2005, EMMCVPR.

[9]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Michael Brady,et al.  Feature-based correspondence: an eigenvector approach , 1992, Image Vis. Comput..

[11]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  Salih O. Duffuaa,et al.  A Linear Programming Approach for the Weighted Graph Matching Problem , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Christoph Schnörr,et al.  Evaluation of Convex Optimization Techniques for the Weighted Graph-Matching Problem in Computer Vision , 2001, DAGM-Symposium.

[14]  Steven Gold,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

[16]  Edwin R. Hancock,et al.  Spectral correspondence for point pattern matching , 2003, Pattern Recognit..

[17]  Bonnie Berger,et al.  Pairwise Global Alignment of Protein Interaction Networks by Matching Neighborhood Topology , 2007, RECOMB.

[18]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .