{
  "abstract": "Scalar rounding decisions interact through matrix multiplication. We study deterministic product-aware rounding after scales, clipping bounds, and grids are fixed, with each active scalar choosing between adjacent levels. For dynamic activation rounding, null-space reduction preserves the relaxed product while leaving at most $r$ fractional decisions, where $r$ is the rank of the active gap-weighted weight block. Conditional-expectation completion gives a deterministic polynomial-time algorithm with squared product error at most $ \\mathrm{OPT}_{\\mathrm{dyn}}+rν_{\\max}^2/4$, where $\\mathrm{OPT}_{\\mathrm{dyn}}$ is the best admissible error and $ν_{\\max}$ is the largest row norm of that block. For reusable static weights, the exact expected product-loss metric is the uncentered input second moment with fixed output bias; free bias recalibration yields the centered covariance. Exact optimization is NP-hard even at rank one. In balanced blocks with $K=1024$ and $r=16$, conditional- expectation completion attains a dither-normalized median error of $0.010$, compared with $0.899$ for round-to-nearest. Clipping-aware initialization reduces median normalized error by a factor of $43.4$ at ten-percent clipping. Held-out Digits experiments show that retaining the input mean or correcting the output bias improves median product error over round-to-nearest in all four tested bit-width and calibration-size settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31641",
  "author_total": 5,
  "authors": [
    {
      "name": "Piyush Sao"
    },
    {
      "name": "Narasinga Miniskar"
    },
    {
      "name": "Pedro Valero-Lara"
    },
    {
      "name": "Keita Teranishi"
    },
    {
      "name": "Sudip Seal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4hqr6fry3n5xp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31641",
  "fulltext_url": null,
  "id": "4hqr6fry3n5xp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31641",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1703|Numerical Methods and Algorithms"
    ]
  ],
  "openalex": "W7214793638",
  "pdf_url": "https://arxiv.org/pdf/2609.31641",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4hqr6fry3n5xp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Product-Aware Deterministic Rounding for Quantized Matrix Multiplication",
  "title_text": "Product-Aware Deterministic Rounding for Quantized Matrix Multiplication",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4hqr6fry3n5xp",
  "venue": "arXiv",
  "versions": []
}