{
  "abstract": "We consider the problem of minimizing error in quantized matrix multiplication $C=AB$. Scalar quantization of the factors introduces rounding errors whose scale depends on the maximum absolute entries -- the ranges -- of their rows and columns. These ranges determine the quantization grid steps. To reduce the error, we optimize over product-preserving transformations that alter the factor ranges and grid steps without changing $C$. Specifically, we seek the smallest leading expected squared error over invertible inner changes of basis and orthogonal outer rotations. Under independent, zero-mean subtractive dither noise on an unbounded lattice, we prove the output-only bound $E_{\\rm lead} \\ge (c_A+c_B)/K \\Vert AB\\Vert_*^2$, where $K$ is the inner dimension, $c_A$ and $c_B$ are normalized noise variances, and $\\Vert AB\\Vert_*$ is the nuclear norm. The bound is tight: an SVD-aligned Hadamard construction attains the infimum whenever a Hadamard matrix of order $K$ exists, including every power of two, while an SVD-aligned DCT construction is within a factor of two for every $K$. Without outer rotations, Gram-matrix balancing minimizes factorization energy, and finite-set flattening achieves the bound within $C\\log(K(m+n))$. For power-of-two $K$, conditional expectations deterministically select the Hadamard signs in $O((m+n)K^2)$ exact-real operations. Synthetic experiments verify both constructions and illustrate the tradeoff between regularization and conditioning. These results characterize the full-gauge optimum and quantify the cost of preserving row and column indices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05641",
  "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/29530800xws0c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05641",
  "fulltext_url": null,
  "id": "29530800xws0c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05641",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212009216",
  "pdf_url": "https://arxiv.org/pdf/2609.05641",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29530800xws0c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products",
  "title_text": "A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/29530800xws0c",
  "venue": "arXiv",
  "versions": []
}