SDP Achieves Exact Minimax Optimality in Phase Synchronization

• URL: http://arxiv.org/abs/2101.02347v1
• Date: Thu, 7 Jan 2021 03:14:05 GMT
• Title: SDP Achieves Exact Minimax Optimality in Phase Synchronization
• Authors: Chao Gao and Anderson Y. Zhang
• Abstract summary: We study the phase synchronization problem with noisy measurements $Y=z*z*+sigma WinmathbbCntimes ntimes n. We prove that SDP achieves error bound $ (1+o)fracnp22np$ under a squared $ell$ loss.
• Score: 19.909352968029584
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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