A Path Following Algorithm for the Graph Matching Problem

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.

[1]  Edwin R. Hancock,et al.  Correspondence matching using kernel principal components analysis and label consistency constraints , 2006, Pattern Recognit..

[2]  Franz Rendl,et al.  QAPLIB – A Quadratic Assignment Problem Library , 1997, J. Glob. Optim..

[3]  RangarajanAnand,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996 .

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

[5]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[6]  James Nga-Kwok Liu,et al.  An oscillatory elastic graph matching model for recognition of offline handwritten Chinese characters , 1999, 1999 Third International Conference on Knowledge-Based Intelligent Information Engineering Systems. Proceedings (Cat. No.99TH8410).

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

[8]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[9]  Harold W. Kuhn,et al.  The Hungarian method for the assignment problem , 1955, 50 Years of Integer Programming.

[10]  丸山 徹 Convex Analysisの二,三の進展について , 1977 .

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

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

[13]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[14]  U. Feige,et al.  Spectral Graph Theory , 2015 .

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

[16]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[17]  室 章治郎 Michael R.Garey/David S.Johnson 著, "COMPUTERS AND INTRACTABILITY A guide to the Theory of NP-Completeness", FREEMAN, A5判変形判, 338+xii, \5,217, 1979 , 1980 .

[18]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[19]  J. Milnor Topology from the differentiable viewpoint , 1965 .

[20]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[21]  S ShapiroLarry,et al.  Feature-based correspondence , 1992 .

[22]  Kurt M. Anstreicher,et al.  A new bound for the quadratic assignment problem based on convex quadratic programming , 2001, Math. Program..

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

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

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

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

[27]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

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

[29]  Erik L. L. Sonnhammer,et al.  Inparanoid: a comprehensive database of eukaryotic orthologs , 2004, Nucleic Acids Res..

[30]  William R Taylor,et al.  Protein Structure Comparison Using Bipartite Graph Matching and Its Application to Protein Structure Classification * , 2002, Molecular & Cellular Proteomics.

[31]  F. Makedon,et al.  A bipartite graph matching framework for finding correspondences between structural elements in two proteins , 2004, The 26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[32]  Adrian S. Lewis,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

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

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

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

[36]  P. Foggia,et al.  Performance evaluation of the VF graph matching algorithm , 1999, Proceedings 10th International Conference on Image Analysis and Processing.

[37]  Leon F. McGinnis,et al.  Implementation and Testing of a Primal-Dual Algorithm for the Assignment Problem , 1983, Oper. Res..

[38]  Edwin R. Hancock,et al.  Alignment and Correspondence Using Singular Value Decomposition , 2000, SSPR/SPR.

[39]  Douglas C. Schmidt,et al.  A Fast Backtracking Algorithm to Test Directed Graphs for Isomorphism Using Distance Matrices , 1976, J. ACM.

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

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

[42]  Fred L. Bookstein,et al.  Principal Warps: Thin-Plate Splines and the Decomposition of Deformations , 1989, IEEE Trans. Pattern Anal. Mach. Intell..