Permutation synchronization is an important problem in computer science that constitutes the key step of many computer vision tasks. The goal is to recover $n$ latent permutations from their noisy and incomplete pairwise measurements. In recent years, spectral methods have gained increasing popularity thanks to their simplicity and computational efficiency. Spectral methods utilize the leading eigenspace $U$ of the data matrix and its block submatrices $U_1,U_2,\ldots, U_n$ to recover the permutations. In this paper, we propose a novel and statistically optimal spectral algorithm. Unlike the existing methods which use $\{U_jU_1^\top\}_{j\geq 2}$, ours constructs an anchor matrix $M$ by aggregating useful information from all the block submatrices and estimates the latent permutations through $\{U_jM^\top\}_{j\geq 1}$. This modification overcomes a crucial limitation of the existing methods caused by the repetitive use of $U_1$ and leads to an improved numerical performance. To establish the optimality of the proposed method, we carry out a fine-grained spectral analysis and obtain a sharp exponential error bound that matches the minimax rate.

翻译：排列同步是计算机科学中的一个重要问题，它构成了许多计算机视觉任务的关键步骤。其目标是从噪声且不完整的成对测量中恢复$n$个潜在排列。近年来，谱方法因其简单性和计算效率而日益受到青睐。谱方法利用数据矩阵的主导特征空间$U$及其块子矩阵$U_1,U_2,\ldots, U_n$来恢复排列。在本文中，我们提出了一种新颖且统计上最优的谱算法。与现有方法使用$\{U_jU_1^\top\}_{j\geq 2}$不同，我们的方法通过聚合所有块子矩阵中的有用信息构建一个锚定矩阵$M$，并通过$\{U_jM^\top\}_{j\geq 1}$估计潜在排列。这一改进克服了现有方法因重复使用$U_1$而导致的关键局限性，并带来了数值性能的提升。为证明所提方法的最优性，我们进行了精细的谱分析，并获得了与极小化极大速率匹配的尖锐指数误差界。