A Path Following Algorithm for the Graph Matching Problem

Abstract

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.

Graduated Consistency-Regularized Optimization for Multi-graph Matching

“…Some recent work first relax the objective function to convex-concave formulation [52,49]. Then the optimal solutions are achieved using the so-called path following strategy and a modified version of the Frank-Wolfe algorithm [49].…”

“…Some recent work first relax the objective function to convex-concave formulation [52,49]. Then the optimal solutions are achieved using the so-called path following strategy and a modified version of the Frank-Wolfe algorithm [49]. Probabilistic matching paradigms are also developed, which have shown unique power in interpreting and addressing hypergraph matching problems [9,28,50].…”

A flexible tensor block coordinate ascent scheme for hypergraph matching

“…hypergraph matching has been used in a variety of problems in computer vision such as object recognition [20], feature correspondences [7,24], shape matching [11,23,25] and surface registration [27]. Given two sets of points, the task is to find the correspondences between them based on extracted features and/or geometric information.…”

“…Although the algorithm comes without theoretical guarantees, it turns out to perform very well in practice, in particular, when one has a large number of outliers. Other second order methods include the work of Zhou and Torre [29] and Zaslavskiy et al [25], where they propose deformable graph matching (DGM) and a path-following algorithm respectively.…”

Real-Time Exact Graph Matching with Application in Human Action Recognition

“…In [19] this is extended to hyper-graphs and the Eigenproblem is solved efficiently with an iterative algorithm. In [25], a convex-concave programming approach is employed on a least-squares problem of the permutation matrices. Several methods decompose the original problem into sub problems which are solved with different optimization tools like graph cuts [20,26].…”