Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

TL;DR

This paper investigates the estimation of orthogonal and rotation group synchronization, proposing an iterative polar decomposition algorithm that achieves minimax optimality in terms of estimation error.

Contribution

It introduces an iterative polar decomposition method for group synchronization that is proven to be minimax optimal, matching the lower bounds for estimation error.

Findings

The proposed algorithm achieves an error of $(1+o(1))rac{\sigma^2 d(d-1)}{2np}$.

A minimax lower bound is established, confirming the optimality of the algorithm.

The method performs well under the probabilistic observation model with Gaussian noise.

Abstract

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.