{
  "abstract": "Witsenhausen's counterexample is a canonical problem in decentralized control: a two-stage linear-quadratic-Gaussian team problem in which nonlinear policies strictly outperform every affine one. This paper develops a variational and computational framework for the scalar counterexample based on optimal transport. Building on the transport formulation of Wu and Verdú, we absorb the first controller into a transport map and recast the problem as $J^\\star=\\inf_Q\\{k^2W_2^2(P,Q)+\\mathrm{mmse}(Q)\\}$, which trades a quadratic Wasserstein transport cost against a minimum mean-square estimation error. The optimal first controller is the monotone rearrangement that pushes the prior $P$ to a minimizer $Q^\\star$. We establish the existence and absolute continuity of minimizers, derive an Euler--Lagrange optimality condition, and obtain a semi-closed-form Gaussian benchmark with an explicit second-variation stability threshold. We then show that restricting to finitely supported laws yields an MMSE-regularized optimal quantizer: its optimal values decrease to $J^\\star$, its weak limit points are minimizers of the continuous problem, and its stationarity conditions pair centroid levels with power cells, recovering Lloyd--Max quantization as control becomes expensive. This structure leads to an estimation-aware Lloyd algorithm with signalling-based initialization. As $k\\to\\infty$, the optimal controllers converge to the classical affine pair; as $k\\to0$, $J^\\star(k)=σ^2k^2+o(k^2)$ and the optimal laws collapse onto $δ_0$, so signalling constellations are a moderate-$k$ phenomenon. Numerical experiments corroborate the theory, and the framework extends to the vector counterexample, where the finite-level program becomes an MMSE-regularized vector quantizer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18992",
  "author_total": 1,
  "authors": [
    {
      "name": "Quanyan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cr0b3r26wc6f1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18992",
  "fulltext_url": null,
  "id": "cr0b3r26wc6f1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18992",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213503576",
  "pdf_url": "https://arxiv.org/pdf/2609.18992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cr0b3r26wc6f1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.18992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "From Optimal Transport to Optimal Quantization: A Variational Study of the Witsenhausen Counterexample",
  "title_text": "From Optimal Transport to Optimal Quantization: A Variational Study of the Witsenhausen Counterexample",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/cr0b3r26wc6f1",
  "venue": "arXiv",
  "versions": []
}