SDP Achieves Exact Minimax Optimality in Phase Synchronization

Abstract

We study the phase synchronization problem with noisy measurements Y = z * z * H + σW ∈ C n×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 bound2np under a normalized squared ℓ 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 Z 2 synchronization, and we achieve the minimax optimal error exp −(1 − o(1)) np 2σ 2 with a sharp constant in the exponent.

Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

“…Our paper is focused on the setting where d does not grow with the sample size n. This covers the most interesting applications in the literature for d = 3, though an extension of our result to a growing d would also be theoretically interesting. We exclude the case d = 1, because SO(1) is a degenerate set, and the problem over O(1) = {−1, 1} is known as Z 2 synchronization, whose minimax rate has already been derived in the literature [11,15]. It is interesting to note that the minimax rate of Z 2 synchronization is exponential instead of the polynomial rate of O(d) synchronization for d ≥ 2.…”