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 |$Z_i^*$| 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 that leads to the optimality of the proposed algorithm as it achieves the exact minimax risk. [ABSTRACT FROM AUTHOR]