SDP Achieves Exact Minimax Optimality in Phase Synchronization

TL;DR

This paper proves that a semidefinite programming relaxation for phase synchronization achieves the minimax optimal error bound, matching the theoretical lower limit, and unifies the analysis of multiple methods for the problem.

Contribution

It establishes the statistical optimality of an SDP relaxation for phase synchronization, with sharp constants, and introduces a unified analysis framework connecting MLE, SDP, and power methods.

Findings

SDP achieves the minimax optimal error bound in phase synchronization.

The analysis unifies the proofs for MLE, SDP, and power iteration methods.

The approach extends to Z2 synchronization with sharp error bounds.

Abstract

We study the phase synchronization problem with noisy measurements $Y=z^*z^{*H}+\sigma W\in\mathbb{C}^{n\times n}$, 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 $Y_{jk}$ is observed with probability $p$. We prove that an SDP relaxation of the MLE achieves the error bound $(1+o(1))\frac{\sigma^2}{2np}$ under a normalized squared $\ell_2$ loss. This result matches the minimax lower bound of the problem, and even the leading constant is sharp. The analysis of the SDP is based on an equivalent non-convex programming whose solution can be characterized as a fixed point of the generalized power iteration lifted to a higher dimensional space. This viewpoint unifies the proofs of the statistical optimality of three different methods: MLE, SDP, and generalized power method. The technique is also applied to the analysis of the SDP for $\mathbb{Z}_2$ synchronization, and we achieve the minimax optimal error $\exp\left(-(1-o(1))\frac{np}{2\sigma^2}\right)$ with a sharp constant in the exponent.