Graph Matching by Simplified Convex-Concave Relaxation Procedure

Abstract

The convex and concave relaxation procedure (CCRP) was recently proposed and exhibited state-of-the-art performance on the graph matching problem. However, CCRP involves explicitly both convex and concave relaxations which typically are difficult to find, and thus greatly limit its practical applications. In this paper we propose a simplified CCRP scheme, which can be proved to realize exactly CCRP, but with a much simpler formulation without needing the concave relaxation in an explicit way, thus significantly simplifying the process of developing CCRP algorithms. The simplified CCRP can be generally applied to any optimizations over the partial permutation matrix, as long as the convex relaxation can be found. Based on two convex relaxations, we obtain two graph matching algorithms defined on adjacency matrix and affinity matrix, respectively. Extensive experimental results witness the simplicity as well as state-of-the-art performance of the two simplified CCRP graph matching algorithms.

Appendix: Proof of Corollary 1

To prove Corollary 1, we prove $-\mathbf {x}^{\top } \mathbf {K} \mathbf {x}=\parallel \mathbf {A}_{D}-\mathbf {X}\mathbf {A}_{M} \mathbf {X}^{\top }\parallel _{F}^2$, where $\mathbf {K}$ is given by (3), and $\mathbf {x}=\mathrm {vec}(\mathbf {X})$. Writing the partial permutation matrix $\mathbf {X}\in \mathbb {R}^{N\times M}$ as

$$\begin{aligned} \mathbf {X}^{\top }=[\mathbf {e}_{\pi (1)},\mathbf {e}_{\pi (2)},\ldots \mathbf {e}_{\pi (N)}] \end{aligned}$$

where $\mathbf {e}_{\pi (i)}$ denotes a column vector of length $M$ with 1 at the position $\pi (i)$ and 0 every other position, it can be then shown that

$$\begin{aligned} -\mathbf {x}^{\top }\mathbf {K}\mathbf {x}&= -\sum _{i=1}^N\sum _{j=1}^N \mathbf {K}_{i\pi (i),j\pi (j)}\nonumber \\&= \sum _{i=1}^N\sum _{j=1}^N({A_D}_{ij}-{A_M}_{\pi (i)\pi (j)})^2 \end{aligned}$$

(30)

On the other hand, the term $\mathbf {X}\mathbf {A}_{M}\mathbf {X}^{\top }$ can be equivalently written as $\{{\mathbf {A}_{M}}_{\pi (i)\pi (j)}\}^{N\times N}$, and consequently,

$$\begin{aligned} \parallel \mathbf {A}_{D}-\mathbf {X}\mathbf {A}_{M}\mathbf {X}^{\top } \parallel _{F}^2&= \parallel \mathbf {A}_{D}-\{{\mathbf {A}_{M}}_{\pi (i)\pi (j)}\}^{N\times N}\parallel _{F}^2\nonumber \\&= \sum _{i=1}^N\sum _{j=1}^N({A_D}_{ij}-{A_M}_{\pi (i)\pi (j)})^2\nonumber \\ \end{aligned}$$

(31)

Thus, $-\mathbf {x}^{\top }\mathbf {K}\mathbf {x}=\parallel \mathbf {A}_{D}-\mathbf {X}\mathbf {A}_{M}\mathbf {X}^{\top } \parallel _{F}^2$, and the proof is accomplished.$\square$