{
  "abstract": "We study gradient flow in a two-layer linear network in the high-dimensional regime where the input, inner and output dimensions diverge proportionally. Under whitened inputs and isotropic imbalance, we combine a closed-form solution of the dynamics with prior random matrix theory results to derive a high-dimensional limit for the dynamics. Our method yields exact asymptotic spectral distributions for the weight matrices at all times and characterizes the evolution of the learned subspace. In the small-initialization limit, we identify a silent alignment phenomenon: the singular directions of the network align on an order-one timescale, while the singular values and empirical loss remain at their initial values. Visible learning occurs later, on a logarithmic timescale, as singular values progressively escape zero and the loss decreases. This phase is characterized by a saddle-to-saddle dynamics, which becomes a continuous process of spectral learning in the high-dimensional limit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29864",
  "author_total": 3,
  "authors": [
    {
      "name": "Simon Martin"
    },
    {
      "name": "Francis Bach"
    },
    {
      "name": "Giulio Biroli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/92y55q24xxgq1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29864",
  "fulltext_url": null,
  "id": "92y55q24xxgq1",
  "issns": [],
  "landing_url": "https://hal.science/hal-05824844v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "arxiv",
      "cs.LG"
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      "arxiv",
      "stat.ML"
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      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
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  "openalex": "W7223192068",
  "pdf_url": "https://hal.science/hal-05824844/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/92y55q24xxgq1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:59Z",
      "native_id": "hal-05824844",
      "source": "hal",
      "url": "https://hal.science/hal-05824844v1"
    },
    {
      "first_seen": "2026-10-30T12:34:29Z",
      "native_id": "2610.29864",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29864"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Silent Alignment in Two-Layer Linear Networks: A High-Dimensional Perspective",
  "title_text": "Silent Alignment in Two-Layer Linear Networks: A High-Dimensional Perspective",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/92y55q24xxgq1",
  "venue": "HAL",
  "versions": []
}