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arXiv · 2109.13491

Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

Chao Gao·Anderson Y. Zhang
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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.

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Chao Gao, Anderson Y. Zhang. 2021-09-28. Optimal Orthogonal Group Synchronization and Rotation Group Synchronization. https://arxiv.org/abs/2109.13491

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