Optimal Orthogonal Group Synchronization and Rotation Group Synchronization
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We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is Y_ij = Z_i^* Z_j^*T + σ W_ij∈ℝ^d× 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))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.
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