SDP Achieves Exact Minimax Optimality in Phase Synchronization

We study the phase synchronization problem with noisy measurements
$Y=z^{*}z^{* { \mathrm {\scriptscriptstyle H} }}+\sigma W\in \mathbb {C}^{n\times n}$
, 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.