{"paper":{"title":"Finding Low-Rank Solutions via Non-Convex Matrix Factorization, Efficiently and Provably","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.IT","cs.LG","cs.NA","math.IT"],"primary_cat":"math.OC","authors_text":"Anastasios Kyrillidis, Constantine Caramanis, Dohyung Park, Sujay Sanghavi","submitted_at":"2016-06-10T03:18:01Z","abstract_excerpt":"A rank-$r$ matrix $X \\in \\mathbb{R}^{m \\times n}$ can be written as a product $U V^\\top$, where $U \\in \\mathbb{R}^{m \\times r}$ and $V \\in \\mathbb{R}^{n \\times r}$. One could exploit this observation in optimization: e.g., consider the minimization of a convex function $f(X)$ over rank-$r$ matrices, where the set of rank-$r$ matrices is modeled via the factorization $UV^\\top$. Though such parameterization reduces the number of variables, and is more computationally efficient (of particular interest is the case $r \\ll \\min\\{m, n\\}$), it comes at a cost: $f(UV^\\top)$ becomes a non-convex functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03168","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":""},"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"}