{
  "abstract": "Procrustes-Wasserstein alignment jointly estimates a matching and rotation without supplied correspondences, but alternating minimization can stop at suboptimal solutions. Rubix solves the equally weighted planar problem globally under squared Euclidean loss. Each matching $σ$ of two centered $n$-point sets defines a complex correlation $z_σ=\\sum_i\\bar x_i y_{σ(i)}$. Their convex hull is the permutation polygon: supporting vertices give optimal matchings at fixed rotations, and the farthest vertex gives the global alignment. We prove the sharp bound of $n(n-1)$ vertices for $n\\ge2$, answering Rote's rotation-assignment open problem. In exact arithmetic, assignment queries recover the polygon in $\\mathcal O(n^5)$ operations. Assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound. On timed MPEG-7 shape pairs, Rubix attains every numerical reference value in 12 ms on average, 50 times faster than a rotation grid at the same accuracy. Its distances improve gravity-aligned matching of real 3D scans, shape retrieval and noisy crystal classification over alternating minimization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10408",
  "author_total": 3,
  "authors": [
    {
      "name": "Subhransu S. Bhattacharjee"
    },
    {
      "name": "Dylan Campbell"
    },
    {
      "name": "Rahul Shome"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dn40vz39e5kq2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10408",
  "fulltext_url": null,
  "id": "dn40vz39e5kq2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10408",
  "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.CG"
    ],
    [
      "arxiv",
      "cs.CV"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.RO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7220978919",
  "pdf_url": "https://arxiv.org/pdf/2610.10408",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dn40vz39e5kq2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10408",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10408"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry",
  "title_text": "Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/dn40vz39e5kq2",
  "venue": "arXiv",
  "versions": []
}