Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

Abstract

We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model iswhere 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)) σ 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.