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 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.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,

Semi-naive dimensional renormalization

We propose a treatment of $\gamma^5$ in dimensional regularization which is based on an algebraically consistent extension of the Breitenlohner-Maison-'t Hooft-Veltman (BMHV) scheme; we define the corresponding minimal renormalization scheme and show its equivalence with a non-minimal BMHV scheme. The restoration of the chiral Ward identities requires the introduction of considerably fewer finite counterterms than in the BMHV scheme. This scheme is the same as the minimal naive dimensional renormalization in the case of diagrams not involving fermionic traces with an odd number of $\gamma^5$, but unlike the latter it is a consistent scheme. As a simple example we apply our minimal subtraction scheme to the Yukawa model at two loops in presence of external gauge fields.Comment: 28 pages, 3 figure