{
  "abstract": "Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity, and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.13612",
  "author_total": 5,
  "authors": [
    {
      "name": "Yatin Dandi"
    },
    {
      "name": "Matteo Vilucchio"
    },
    {
      "name": "Luca Arnaboldi"
    },
    {
      "name": "Hugo Tabanelli"
    },
    {
      "name": "Florent Krzakala"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ds8ns1vn7ex68.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.13612",
  "fulltext_url": null,
  "id": "ds8ns1vn7ex68",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.13612",
  "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",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7161173263",
  "pdf_url": "https://arxiv.org/pdf/2605.13612",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ds8ns1vn7ex68.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2605.13612",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.13612"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning",
  "title_text": "Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:25Z",
  "url": "https://research.draconitic.com/paper/ds8ns1vn7ex68",
  "venue": "arXiv",
  "versions": []
}