{
  "abstract": "We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector \\(x\\) of active features is encoded through an overcomplete dictionary \\(W\\), and feature recovery is performed by applying \\(\\operatorname{ReLU}(W^\\top W x+b)\\) with an appropriate bias vector \\(b\\). We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order \\(d/\\log n\\). In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with \\(n>d+1\\), we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization---which should be of independent interest to frame theorists---of the distribution of signs in the Gram matrix.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27540",
  "author_total": 4,
  "authors": [
    {
      "name": "Michael I. Ivanitskiy"
    },
    {
      "name": "John Jasper"
    },
    {
      "name": "Emily J. King"
    },
    {
      "name": "Dustin G. Mixon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4kfgpbg4501vs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27540",
  "fulltext_url": null,
  "id": "4kfgpbg4501vs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27540",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7204756987",
  "pdf_url": "https://arxiv.org/pdf/2608.27540",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4kfgpbg4501vs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27540",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27540"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Towards a mathematical theory of superposition",
  "title_text": "Towards a mathematical theory of superposition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/4kfgpbg4501vs",
  "venue": "arXiv",
  "versions": []
}