{"paper":{"title":"Benign landscapes for synchronization on spheres via normalized Laplacian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.ST","stat.TH"],"primary_cat":"math.OC","authors_text":"Andrew D. McRae","submitted_at":"2025-03-24T15:47:19Z","abstract_excerpt":"We study the nonconvex optimization landscapes of synchronization problems on spheres. First, we present new results for the statistical problem of synchronization over the two-element group $\\mathbf{Z}_2$. We consider the nonconvex least-squares problem with $\\mathbf{Z}_2 = \\{\\pm 1\\}$ relaxed to the unit sphere in $\\mathbf{R}^r$ for $r \\geq 2$; for several popular models, including graph clustering under the binary stochastic block model, we show that, for any $r \\geq 2$, every second-order critical point recovers the ground truth in the asymptotic regimes where exact recovery is information-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.18801","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.18801/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}