{
  "abstract": "We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss--Newton (RGN) methods. For an $n\\times n$ matrix of rank $r$ with incoherence parameter $μ$ and condition number $κ$, the two methods achieve exact recovery with high probability from $O(μnr\\log n\\log(nκ))$ and $O(μnr\\log n\\log(2μrκ))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17048",
  "author_total": 3,
  "authors": [
    {
      "name": "Jian-Feng Cai"
    },
    {
      "name": "Xiliang Lu"
    },
    {
      "name": "Juntao You"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fvz49a2wbk9et.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17048",
  "fulltext_url": null,
  "id": "fvz49a2wbk9et",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17048",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7213423063",
  "pdf_url": "https://arxiv.org/pdf/2609.17048",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fvz49a2wbk9et.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17048",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17048"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Near-Optimal Nonconvex Matrix Completion",
  "title_text": "Near-Optimal Nonconvex Matrix Completion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/fvz49a2wbk9et",
  "venue": "arXiv",
  "versions": []
}