{
  "abstract": "We study the generalization of spectral gradient descent (SpecGD) in overparameterized matrix classification with corrupted labels. Each input combines a shared low-rank signal with a rank-one sample-specific perturbation, referred to as a shortcut, that enables memorization but does not generalize. We contrast collapsed shortcuts, which share a singular direction, with dispersed shortcuts, which occupy distinct singular directions. Changing only this geometry can reverse the relative generalization of GD and SpecGD: collapsed shortcuts can favor SpecGD, while dispersed shortcuts can favor GD. In the dispersed regime, exact shortcut orthogonality eliminates the signal from the late-stage SpecGD direction, while vanishing random correlations collectively generate a small but generalization-relevant signal through a second-order effect. To identify the direction selected by SpecGD, which the spectral max-margin problem alone does not determine, we combine a refined analysis of its dual with the exponentiated-gradient dynamics of normalized loss weights. Finally, we show that a single SpecGD step can already interpolate and generalize well, while continued training converges to a direction with substantially worse generalization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32270",
  "author_total": 3,
  "authors": [
    {
      "name": "Guillaume Braun"
    },
    {
      "name": "Ichiro Hashimoto"
    },
    {
      "name": "Masaaki Imaizumi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n4h48a8n6zsc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32270",
  "fulltext_url": null,
  "id": "2n4h48a8n6zsc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32270",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214746260",
  "pdf_url": "https://arxiv.org/pdf/2609.32270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n4h48a8n6zsc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:26Z",
      "native_id": "2609.32270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32270"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Overfitting of Spectral Gradient Descent: How Matrix Geometry shapes Generalization and Implicit Bias",
  "title_text": "Overfitting of Spectral Gradient Descent: How Matrix Geometry shapes Generalization and Implicit Bias",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/2n4h48a8n6zsc",
  "venue": "arXiv",
  "versions": []
}