A Path Following Algorithm for the Graph Matching Problem

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 31; no. 12; pp. 2227 - 2242
• Main Authors: Zaslavskiy, M., Bach, F., Vert, J.-P.
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.12.2009; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Artificial Intelligence; Bioinformatics; classification; Combinatorial analysis; Combinatorics; Combinatorics. Ordered structures; Computer Science; Computer science; control theory; systems; convex programming; Data Structures and Algorithms; Exact sciences and technology; Gradient methods; Graph algorithms; Graph matching; Graph theory; Humans; Image Processing; Image Processing, Computer-Assisted; Information retrieval. Graph; Interpolation; Life Sciences; Machine Learning; Machine learning algorithms; Mathematical analysis; Mathematics; Matrices; Optimization; Optimization methods; Pattern Recognition, Automated - statistics & numerical data; Pattern recognition. Digital image processing. Computational geometry; Programming; Proteins; Quadratic programming; Quantitative Methods; Retina; Retinal Vessels - anatomy & histology; Sciences and techniques of general use; Stochastic processes; Studies; Theoretical computing; Gradient methods; Artificial Intelligence; Computing Methodologies; Object recognition; Convex programming; Optimization; Discrete Mathematics; Constrained optimization; Learning; Scene Analysis; Pattern Recognition; Graph Theory; Mathematics of Computing; Numerical Analysis; Machine learning; Image Processing and Computer Vision; Graph algorithms; Quadratic programming methods; Image processing; Combinatorial problem; Competitiveness; Path following; Retina; Combinatorial optimization; Least squares method; Graph transformation; Classification; Pattern analysis; Gradient method; gradient methods; Mathematical programming; Linear interpolation; Probabilistic approach; Permutation; Rewriting; Graph theory; machine learning; Weighted graph; Search algorithm; Ideogram; Graph method; Manuscript character; Artificial intelligence; Graph matching
• ISSN: 0162-8828; 1939-3539; 1939-3539
• DOI: 10.1109/TPAMI.2008.245

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.