A Novel and Optimal Spectral Method for Permutation Synchronization
This work addresses a key step in computer vision tasks, offering a statistically optimal solution with improved performance, though it is incremental as it builds on existing spectral methods.
The authors tackled the problem of permutation synchronization, which aims to recover latent permutations from noisy pairwise measurements, by proposing a novel spectral algorithm that aggregates information from all block submatrices to overcome limitations of existing methods, achieving a sharp exponential error bound that matches the minimax rate.
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 of 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.