{"paper":{"title":"Guarantees of Riemannian Optimization for Low Rank Matrix Completion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.NA","authors_text":"Jian-Feng Cai, Ke Wei, Shingyu Leung, Tony F. Chan","submitted_at":"2016-03-21T20:51:16Z","abstract_excerpt":"We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\\times n$ rank $r$ matrix are sampled independently and uniformly with replacement. We first prove that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided \\begin{align*} m\\geq C_\\kappa n^{1.5}r\\log^{1.5}(n), \\end{align*"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06610","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"}