{
  "abstract": "We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm--regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.",
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  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06146",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew D Mcrae"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33a45he97s9rh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06146",
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  "id": "33a45he97s9rh",
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  "landing_url": "https://hal.science/hal-05740523v1",
  "language": "en",
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  "metadata_licence": "CC0-1.0",
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  "pdf_url": "https://hal.science/hal-05740523/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33a45he97s9rh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:29Z",
      "native_id": "hal-05740523",
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      "url": "https://hal.science/hal-05740523v1"
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    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06146",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06146"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements",
  "title_text": "Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/33a45he97s9rh",
  "venue": "HAL",
  "versions": []
}