{
  "abstract": "Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24041",
  "author_total": 4,
  "authors": [
    {
      "name": "Kevin Han Huang"
    },
    {
      "name": "Haoyu Ye"
    },
    {
      "name": "Somak Laha"
    },
    {
      "name": "Morgane Austern"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4dc6nkdb3esk3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24041",
  "fulltext_url": null,
  "id": "4dc6nkdb3esk3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24041",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7171486155",
  "pdf_url": "https://arxiv.org/pdf/2607.24041",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4dc6nkdb3esk3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2607.24041",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24041"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression",
  "title_text": "The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:42Z",
  "url": "https://research.draconitic.com/paper/4dc6nkdb3esk3",
  "venue": "arXiv",
  "versions": []
}