# Exact Minimax Estimation for Phase Synchronization

arXiv: https://arxiv.org/abs/2010.04345 · HTML: https://mlsanity.com/paper/2010.04345/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2020.

## Built on by (most cited first, 5 of 5)

- [Localization in 1D non-parametric latent space models from pairwise affinities](https://mlsanity.com/paper/2108.03098/) (2021)
- [Optimal Orthogonal Group Synchronization and Rotation Group Synchronization](https://mlsanity.com/paper/2109.13491/) (2021)
- [Orthogonal Group Synchronization with Incomplete Measurements: Error Bounds and Linear Convergence of the Generalized Power Method](https://mlsanity.com/paper/2112.06556/) (2021)
- [SDP Achieves Exact Minimax Optimality in Phase Synchronization](https://mlsanity.com/paper/2101.02347/) (2021)
- [Sharp analysis of EM for learning mixtures of pairwise differences](https://mlsanity.com/paper/2302.10066/) (2023)

## Abstract (arXiv)

We study the phase synchronization problem with measurements Y=z^*z*H+σ W∈mathbbCn× n, where z^* is an n-dimensional complex unit-modulus vector and W is a complex-valued Gaussian random matrix. It is assumed that each entry Yjk is observed with probability p. We prove that the minimax lower bound of estimating z^* under the squared ℓ2 loss is (1-o(1))fracσ22p. We also show that both generalized power method and maximum likelihood estimator achieve the error bound (1+o(1))fracσ22p. Thus, fracσ22p is the exact asymptotic minimax error of the problem. Our upper bound analysis involves a precise characterization of the statistical property of the power iteration. The lower bound is derived through an application of van Trees' inequality.