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 Yij = Zi^* Zj*T + σ Wij∈ℝd× d where Wij is a Gaussian random matrix and Zi^* is either an orthogonal matrix or a rotation matrix, and each Yij 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))fracσ2 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.