{"paper":{"title":"Provable Acceleration of Nesterov's Accelerated Gradient for Rectangular Matrix Factorization and Linear Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Molei Tao, Rachel Ward, Tuo Zhao, Yuqing Wang, Zhenghao Xu","submitted_at":"2024-10-12T20:33:37Z","abstract_excerpt":"We study the convergence rate of first-order methods for rectangular matrix factorization, which is a canonical nonconvex optimization problem. Specifically, given a rank-$r$ matrix $\\mathbf{A}\\in\\mathbb{R}^{m\\times n}$, we prove that gradient descent (GD) can find a pair of $\\epsilon$-optimal solutions $\\mathbf{X}_T\\in\\mathbb{R}^{m\\times d}$ and $\\mathbf{Y}_T\\in\\mathbb{R}^{n\\times d}$, where $d\\geq r$, satisfying $\\lVert\\mathbf{X}_T\\mathbf{Y}_T^\\top-\\mathbf{A}\\rVert_\\mathrm{F}\\leq\\epsilon\\lVert\\mathbf{A}\\rVert_\\mathrm{F}$ in $T=O(\\kappa^2\\log\\frac{1}{\\epsilon})$ iterations with high probabili"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.09640","kind":"arxiv","version":3},"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/2410.09640/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"}