This paper addresses the Graph Matching problem, which consists of finding the best possible alignment between two input graphs, and has many applications in computer vision, network deanonymization and protein alignment. A common approach to tackle this problem is through convex relaxations of the NP-hard \emph{Quadratic Assignment Problem} (QAP). Here, we introduce a new convex relaxation onto the unit simplex and develop an efficient mirror descent scheme with closed-form iterations for solving this problem. Under the correlated Gaussian Wigner model, we show that the simplex relaxation admits a unique solution with high probability. In the noiseless case, this is shown to imply exact recovery of the ground truth permutation. Additionally, we establish a novel sufficiency condition for the input matrix in standard greedy rounding methods, which is less restrictive than the commonly used `diagonal dominance' condition. We use this condition to show exact one-step recovery of the ground truth (holding almost surely) via the mirror descent scheme, in the noiseless setting. We also use this condition to obtain significantly improved conditions for the GRAMPA algorithm [Fan et al. 2019] in the noiseless setting.

翻译：本文研究图匹配问题，该问题旨在寻找两个输入图之间的最优对齐方案，在计算机视觉、网络去匿名化及蛋白质比对等领域具有广泛应用。解决该问题的常见方法是通过对NP难问题——二次分配问题（QAP）进行凸松弛。本文提出一种新的单位单纯形凸松弛方法，并开发了一种具有闭式迭代的高效镜像下降求解方案。在高斯维格纳相关模型下，我们证明该单纯形松弛解以高概率存在唯一性。在无噪声场景中，该性质可推导出对真实排列的精确恢复。此外，我们为标准贪心舍入法建立了新颖的输入矩阵充分性条件，该条件比常用的"对角占优"条件限制更弱。利用该条件，我们证明了在无噪声设置下，通过镜像下降方案能以几乎必然的概率实现真实排列的单步精确恢复。同时，基于该条件，我们在无噪声设置下为GRAMPA算法[Fan et al. 2019]获得了显著改进的收敛条件。