{"paper":{"title":"Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.LG","cs.NA","math.IT","math.OC"],"primary_cat":"stat.ML","authors_text":"Cong Ma, Jianqing Fan, Yuejie Chi, Yuxin Chen","submitted_at":"2018-03-21T03:14:16Z","abstract_excerpt":"This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest $\\mathbf{x}^{\\natural}\\in\\mathbb{R}^{n}$ from $m$ quadratic equations/samples $y_{i}=(\\mathbf{a}_{i}^{\\top}\\mathbf{x}^{\\natural})^{2}$, $1\\leq i\\leq m$. This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning.\n  We investigate the efficiency of gradient descent (or Wirtinger flow) designed for the nonconvex least squares problem. We prove that under Gaussian designs, gradient descent --- when randomly initialized --- "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07726","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}