{
  "abstract": "GPTQ is Babai's nearest-plane algorithm on the lattice of the Cholesky factor of the layer's input second moment $Σ$. In the high-rate, entropy-coded regime it needs $\\tfrac12\\log_2(\\AM/\\GM)$ of its Cholesky pivots more bits than WaterSIC, which is within $0.255$ bit of the rate-distortion limit for every $Σ$; random rotations close most of this gap in practice, and Ordentlich and Polyanskiy asked for the worst-case gap between rotated GPTQ and WaterSIC. We show that after a Haar rotation the $k$-th pivot converges almost surely, uniformly in $k\\geδn$ and over spectra $ν$ of bounded condition number, to $1/S_ν(-k/n)$, a classical formula from the theory of Lyapunov exponents, and use it to answer the question. Over spectra with condition number $κ$ the worst rotated gap is $\\tfrac12\\log_2κ-\\log_2\\lnκ+\\tfrac12-\\tfrac1{\\ln2}+o(1)$ bits, attained only by two-point spectra, just $\\tfrac12\\log_2\\lnκ+O(1)$ bits below the worst case without rotation, so rotation cannot replace waterfilling in general. It nearly can on real layers: rotation cuts the cost of an $η$-fraction of outlier eigenvalues of size $μ$ from about $ημ$ to $η^2μ/2$ to leading order, but cannot remove spectral spread on a logarithmic scale; for non-Gaussian weights it also adds a weight-side rate cost, which we characterize exactly under a standard high-rate assumption. On $96$ Hessians of Qwen3 and Llama-3.1 the rotated gap has median $0.04$ bit and maximum $0.36$ bit, against $2$--$5$ bits in the worst case for the same condition numbers, and the limit matches it to within $0.009$ bit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01669",
  "author_total": 1,
  "authors": [
    {
      "name": "Runguo Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bbzb01nrrrnr3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01669",
  "fulltext_url": null,
  "id": "bbzb01nrrrnr3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01669",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01669",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bbzb01nrrrnr3.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01669",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01669"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "How Much Does Random Rotation Help GPTQ? Sharp Worst-Case Rate Gaps in the High-Rate Regime",
  "title_text": "How Much Does Random Rotation Help GPTQ? Sharp Worst-Case Rate Gaps in the High-Rate Regime",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/bbzb01nrrrnr3",
  "venue": "arXiv",
  "versions": []
}