{
  "abstract": "Feature learning is widely regarded as the key mechanism distinguishing neural networks from fixed-kernel methods, yet its impact on the induced function space remains poorly understood. In this work, we precisely characterize how the function space spanned by the features of a two-layer neural network evolves during gradient descent training. We prove that, in the highdimensional proportional regime, after a large gradient step the post-update feature distribution is well approximated by a target-dependent spiked Gaussian covariance. This induces a data adaptive kernel that reshapes the function space and modifies its spectral structure. Our analysis reveals that feature learning can be interpreted as a distributional transformation in either parameter space or input space, equivalently as the introduction of a target-dependent kernel. In particular, it selectively amplifies eigenvalues aligned with the target direction and mixes leading eigenfunctions, coupling the top radial mode with a target-aligned quadratic harmonic. Overall, our results provide a precise function-space perspective on early-stage feature learning: rather than just rescaling a fixed kernel, gradient descent induces a data-adaptive deformation that preferentially enhances directions aligned with the signal in the data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.17718",
  "author_total": 4,
  "authors": [
    {
      "name": "João Lobo"
    },
    {
      "name": "Bruno Loureiro"
    },
    {
      "name": "Long Tran-Than"
    },
    {
      "name": "Fanghui Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2y03hfkqmm8cf.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "2y03hfkqmm8cf",
  "issns": [],
  "landing_url": "https://hal.science/hal-05761651v1",
  "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": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05761651/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2y03hfkqmm8cf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:39Z",
      "native_id": "hal-05761651",
      "source": "hal",
      "url": "https://hal.science/hal-05761651v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "How does feature learning reshape the function space?",
  "title_text": "How does feature learning reshape the function space?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:39Z",
  "url": "https://research.draconitic.com/paper/2y03hfkqmm8cf",
  "venue": "HAL",
  "versions": []
}