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.
Categories: math.ST math.OC stat.ML stat.TH