{
  "abstract": "The information bottleneck (IB) seeks a representation $U$ of a source $X$ that retains as much information as possible about a target $Y$, subject to a constraint on $I(U;X)$. A classical argument shows that it suffices to consider representations with at most $|\\mathcal{X}|+1$ symbols, and this bound is known to be tight whenever $|\\mathcal{X}| \\geq 3$. We show that the binary case behaves differently: if $X$ is binary and $Y$ is finite, then for every joint distribution of $(X,Y)$ and every rate constraint, the IB optimum is attained by a binary $U$. Hence the bound $|\\mathcal{U}| \\leq |\\mathcal{X}|+1$ sharpens to $|\\mathcal{U}| \\leq |\\mathcal{X}|$ for binary sources. The proof combines a separating hyperplane argument with the observation that, for a binary source, the ratio of the second derivatives of the two entropy functions involved is concave.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06627",
  "author_total": 2,
  "authors": [
    {
      "name": "Dier Tang"
    },
    {
      "name": "Jun Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77et1yeyy0yqd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06627",
  "fulltext_url": null,
  "id": "77et1yeyy0yqd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06627",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7220697265",
  "pdf_url": "https://arxiv.org/pdf/2610.06627",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77et1yeyy0yqd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06627",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06627"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Cardinality of Optimal Representations in the Binary-Source Information Bottleneck",
  "title_text": "On the Cardinality of Optimal Representations in the Binary-Source Information Bottleneck",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/77et1yeyy0yqd",
  "venue": "arXiv",
  "versions": []
}