Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $Y_{ij} = Z_i^* Z_j^{*T} + \sigma W_{ij}\in$\mathbb{R}$^{d\times d}$where$W_{ij}$ is a Gaussian random matrix and $Z_i^{\ast }$ is either an orthogonal matrix or a rotation matrix, and each $Y_{ij}$ is observed independently with probability $p$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^{\ast }$ and show it has an error of $(1+o(1))\frac{{\sigma }^{2}d(d-1)}{2np}$ when initialized by spectral methods. A matching minimax lower bound is further established which leads to the optimality of the proposed algorithm as it achieves the exact minimax risk.