  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 $Z_i^*$ 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 $Z^*$
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.
