{
  "abstract": "We present an interpretation of the fast matrix multiplication algorithms of Coppersmith and Winograd as aggregations of tensor contractions. This point of view provides a direct connection to older constructions by Pan and to more recent work on bilinear algorithms for symmetric tensor contraction. Further, we provide a simple explicit approach for exact aggregation of tensor contractions, each one yielding an independent matrix-matrix product. These constructions and border rank adaptations thereof do not come close to improving the best known bounds on $ω$ (the matrix multiplication exponent), but provide explicit moderate-size fast matrix multiplication algorithms that achieve a better exponent than Strassen's algorithm and have potential practical value. These finite constructions improve the multiplication-count bounds of the reference algorithms compared here for selected numbers and dimensions of independent square matrix products. In particular, using $\\langle m,m,m\\rangle$ to denote the matrix multiplication tensor for matrices of dimension $m$, we obtain constructions that give bounds such as the border rank bound \\[ \\underline R\\!\\left(\\langle n^2,n^2,n^2\\rangle^{\\oplus9}\\right) \\le (n+1)^6+6(n+1)^5, \\] and the exact rank bound \\[ R\\!\\left(\\langle n^3,n^3,n^3\\rangle^{\\oplus14}\\right)\\le (n+1)^9. \\] We quantify the accuracy of such constructions in finite precision, and provide border-rank constructions that reduce cost with limiting retained fractions of accuracy at least $1/2$, $1/3$, or $1/4$ relative to the classical matrix-multiplication approach under the stated error model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26563",
  "author_total": 2,
  "authors": [
    {
      "name": "Edgar Solomonik"
    },
    {
      "name": "Markus Bläser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evq9s5emfdjjc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26563",
  "fulltext_url": null,
  "id": "evq9s5emfdjjc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26563",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223061070",
  "pdf_url": "https://arxiv.org/pdf/2610.26563",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evq9s5emfdjjc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.26563",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26563"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Simultaneous Matrix Multiplication by Tensor Contraction Aggregation",
  "title_text": "Simultaneous Matrix Multiplication by Tensor Contraction Aggregation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/evq9s5emfdjjc",
  "venue": "arXiv",
  "versions": []
}