{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:23:35Z","timestamp":1787340215507,"version":"build-2736575974"},"reference-count":39,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We study the difference between the maximum likelihood estimation (MLE) and its semidefinite programming (SDP) relaxation for the phase synchronization problem, where [Formula: see text] latent phases are estimated based on pairwise observations corrupted by Gaussian noise at a level [Formula: see text]. While previous studies have established that SDP coincides with the MLE when [Formula: see text], the behavior in the high-noise regime [Formula: see text] remains unclear. We address this gap by quantifying the deviation between the SDP and the MLE in the high-noise regime as [Formula: see text], indicating an exponentially small discrepancy. In fact, we establish more general results for the Burer\u2013Monteiro 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 [Formula: see text] is small. To obtain our results, we develop a refined entrywise analysis of the MLE that is beyond the existing [Formula: see text] analysis in the literature.<\/jats:p>","DOI":"10.1137\/24m1689429","type":"journal-article","created":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T08:07:52Z","timestamp":1770019672000},"page":"60-89","source":"Crossref","is-referenced-by-count":0,"title":["Tightness of SDP and Burer\u2013Monteiro Factorization for Phase Synchronization in a High-Noise Regime"],"prefix":"10.1137","volume":"36","author":[{"given":"Anderson Ye","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, University of Pennsylvania, Philadelphia, PA 19104 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,2,2]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"E. Abbe, L. Massouli\u00e9, A. Montanari, A. Sly, and N. 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D. McRae, Benign Landscapes for Synchronization on Spheres via Normalized Laplacian Matrices, preprint, arXiv:2503.18801, 2025."},{"key":"ref27","first-page":"iaaf012","volume":"14","author":"McRae A. D.","year":"2025","journal-title":"Inf. Inference J IMA."},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1137\/23M1584642"},{"key":"ref29","unstructured":"S. Mei, T. Misiakiewicz, A. Montanari, and R. I. Oliveira, Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality, in Proceedings of the 2017 Conference on Learning Theory, PMLR, 2017, pp. 1476\u20131515."},{"key":"ref30","doi-asserted-by":"crossref","unstructured":"M. Rudelson and R. 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