{
  "abstract": "We introduce a geometric formulation of statistical feature learning for supervised regression. Feature learning is defined through a base--fiber decomposition: the base is the feature-side geometry produced by training, and the fiber is the learned feature space where estimation is performed. We prove this property for spherical mean-field Langevin dynamics, viewed as the Wasserstein gradient flow of a negative entropy-regularized empirical risk. In Gaussian multi-index models, the low-temperature stationary distribution concentrates near the hidden indices, forms a multi-spike structure, and yields parameter recovery with high probability, even though negative entropy regularization penalizes concentration. This concentration has a sharp transition at temperature $λ\\asymp 1$. In Gaussian single-index models, the stationary measure satisfies a concentration property, with parity determining whether it lives on $S_2^{d-1}$ or $\\mathbb{RP}^{d-1}$. The induced learned feature space aligns the regression signal and yields rates $d/N$ and $Md/N$, up to logarithmic factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.31429",
  "author_total": 4,
  "authors": [
    {
      "name": "Zong Shang"
    },
    {
      "name": "Tomoya Wakayama"
    },
    {
      "name": "Guillaume Lecué"
    },
    {
      "name": "Taiji Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0jn534z8w0xh6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.31429",
  "fulltext_url": null,
  "id": "0jn534z8w0xh6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.31429",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7166880049",
  "pdf_url": "https://arxiv.org/pdf/2606.31429",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0jn534z8w0xh6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2606.31429",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.31429"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The Geometry of Statistical Feature Learning in Mean-Field Langevin Dynamics",
  "title_text": "The Geometry of Statistical Feature Learning in Mean-Field Langevin Dynamics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/0jn534z8w0xh6",
  "venue": "arXiv",
  "versions": []
}