IEEE Transactions on Neural Networks (in press) A Lagrangian Relaxation Network for Graph Matching Anand Rangarajan Departments of Diagnostic Radiology and Computer Science Yale University New Haven, CT 06520-8042 e-mail: anand@noodle.med.yale.edu Eric Mjolsness Department of Computer Science and Engineering University of California San Diego (UCSD) La Jolla, CA 92093-0114 e-mail: emj@cs.ucsd.edu Abstract A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs and , find a permutation matrix that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two “solutions.” Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetry-breaking self-amplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and self-amplification terms, we obtain a Lagrangian relaxation network. The network performs min- imization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variables. Simulation results are shown on 100 node random graphs and for a wide range of connectivities. Index Terms: Deterministic annealing, free energy, permutation matrix, Lagrangian decomposition, al- gebraic transformation, Lagrangian relaxation, graph matching, self-amplification, symmetry-breaking, merit function. 1 Introduction Graph matching is an important part of object recognition systems in computer vision. The problem of matching structural descriptions of an object to those of a model is posed as weighted graph matching [1]-[26]. Good, approximate solutions to inexact, weighted graph matching are usually required. Neural network approaches to graph matching [6, 7, 8, 9, 11, 12, 28, 15, 23, 21, 25, 27] share the feature of other recent approaches to graph matching [29, 30, 14, 31, 20, 24] in that they are not restricted to looking for the right isomorphism. Instead, a measure of distance between the two graphs [32] is minimized. The focus shifts from brittle, symbolic, subgraph isomorphism distance metrics [13, 33] to an “energy” or 1