Tightness of SDP and Burer-Monteiro Factorization for Phase Synchronization in High-Noise Regime

We study the difference between the maximum likelihood estimation (MLE) and its semi-definite programming (SDP) relaxation for the phase synchronization problem, where $n$ latent phases are estimated based on pairwise observations corrupted by Gaussian noise at a level $\sigma$. While previous studies have established that SDP coincides with the MLE when $\sigma \lesssim \sqrt{n / \log n}$, the behavior in the high-noise regime $\sigma \gtrsim \sqrt{n / \log n}$ remains unclear. We address this gap by quantifying the deviation between the SDP and the MLE in the high-noise regime as $\exp(-c \frac{n}{\sigma^2})$, indicating an exponentially small discrepancy. In fact, we establish more general results for the Burer-Monteiro (BM) factorization that covers the SDP as a special case: it has the exponentially small deviation from the MLE in the high-noise regime and coincides with the MLE when $\sigma$ is small. To obtain our results, we develop a refined entrywise analysis of the MLE that is beyond the existing $\ell_\infty$ analysis in literature.