On the Convergence of Graph Matching: Graduated Assignment Revisited

We focus on the problem of graph matching that is fundamental in computer vision and machine learning. Many state-of-the-arts frequently formulate it as integer quadratic programming, which incorporates both unary and second-order terms. This formulation is in general NP-hard thus obtaining an exact solution is computationally intractable. Therefore most algorithms seek the approximate optimum by relaxing techniques. This paper commences with the finding of the “

circular

” character of solution chain obtained by the iterative

Gradient Assignment

(via Hungarian method) in the discrete domain, and proposes a method for guiding the solver converging to a fixed point, resulting a convergent algorithm for graph matching in discrete domain. Furthermore, we extend the algorithms to their counterparts in continuous domain, proving the classical graduated assignment algorithm will converge to a double-circular solution chain, and the proposed Soft Constrained Graduated Assignment (SCGA) method will converge to a fixed (discrete) point, both under wild conditions. Competitive performances are reported in both synthetic and real experiments.