{
  "abstract": "We study the mean-squared error (MSE) rate-distortion function (RDF) of finite Gaussian mixture models (GMMs) in $\\mathbb{R}^d$, for which neither the RDF nor an optimal test channel is known in closed form. It can be sandwiched between the genie-aided conditional RDF $R_{\\mathbf{X} | T}(D)$, which can be characterized in closed-form for a GMM, and $R_{\\mathbf{X} | T}(D)+I(\\mathbf{X};T)$, where $T$ denotes the latent component label. We show that the upper bound can be achieved operationally by the proposed Gibbs-relabelling method (GRM), which conveys a label from the posterior $P_{T|\\mathbf{X}}$ through \\textit{channel simulation} at rate $I(\\mathbf{X};T)$, followed by vector quantization of the resulting exactly Gaussian conditional sources by reverse water-filling. Using the Shannon lower bound (SLB), we prove that GRM attains the SLB, and hence the RDF, of a general finite mixture if and only if every component attains its own SLB. For a GMM this holds throughout the fully active interval $D/d<\\min_{k,m}λ_{k,m}$, where $λ_{k,m}$ are the eigenvalues of the component covariances, and for scalar GMMs the condition is also necessary. Since GRM cannot operate below the label cost $I(\\mathbf{X};T)$, we propose a complementary stochastic scalar quantization (SSQ) scheme for low rates, which simulates an $M$-ary reconstruction index. We show that at low rates SSQ with $M\\geq 2$ matches the Shannon RDF to first order: $R(D_{\\max}-δ)=δ/(2τ^2)+o(δ)$, where $τ^2$ is a directional sub-Gaussian proxy of the mixture, and quantify the price of a finite codebook through a strict rate gap at high rates and a distortion floor whose joint scaling in $M$ and number of mixture components is characterized. Numerical results illustrate both regimes and the finite-blocklength cost of channel simulation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15549",
  "author_total": 2,
  "authors": [
    {
      "name": "Weichen Wang"
    },
    {
      "name": "Deniz Gündüz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b35xsthhxhgte.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15549",
  "fulltext_url": null,
  "id": "b35xsthhxhgte",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15549",
  "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",
      "math.IT"
    ],
    [
      "openalex",
      "1707|Advanced Data Compression Techniques"
    ]
  ],
  "openalex": "W7221785815",
  "pdf_url": "https://arxiv.org/pdf/2610.15549",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b35xsthhxhgte.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15549",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15549"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Rate-Distortion Theory of Gaussian Mixtures via Posterior Label Sampling",
  "title_text": "Rate-Distortion Theory of Gaussian Mixtures via Posterior Label Sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/b35xsthhxhgte",
  "venue": "arXiv",
  "versions": []
}