Optimal orthogonal group synchronization and rotation group synchronization

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