{
  "abstract": "Essential Data Points (EDPs) - the vertices of the convex hull of a bilinear data matrix $\\mathbf{D}$ in its row space, column space, or both - are widely used in chemometrics to reduce the size of large data sets while nominally preserving their underlying geometric structure. Using simulated three- and two-component chromatographic/spectral data sets and a real source-apportionment data set ($\\mathbf{D = C\\, A^\\mathsf{T}}$), together with Borgen-Rajkó plots, Procrustes analysis, and variance-covariance comparisons, we show that this preservation is only \\emph{partial}: row-wise EDP reduction preserves the row-space geometry (inner and outer polygons) exactly while distorting the column-space geometry, and column-wise reduction shows the opposite behavior; joint row-and-column reduction distorts both. We then give a rigorous, general proof - based on the four fundamental subspaces of a matrix and its singular value decomposition $\\mathbf{D = U\\,S\\,V^\\mathsf{T}}$ - that the subspace which is \\emph{not} being reduced is always preserved exactly, up to an orthogonal rotation, whereas the subspace whose ambient dimension shrinks is related to the original only through a general, non-orthogonal isomorphism. This distinction is confirmed numerically to machine precision ($\\sim 10^{-14}$-$10^{-16}$) on the real data set, and a deliberate negative control confirms that the \"ambient-shrinking\" map is genuinely non-orthogonal (residual $\\approx 1$). These results demonstrate that the apparent rotation of an EDP-reduced polygon relative to the original is not, in general, a rigid rotation, and that visual or numerical comparisons between an EDP-reduced data set and the original data require an explicit, mode-dependent change-of-basis correction before any geometric or statistical conclusion can be drawn. A MATLAB implementation of this correction is provided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25377",
  "author_total": 3,
  "authors": [
    {
      "name": "Hamideh Bakhshi"
    },
    {
      "name": "Hamid Abdollahi"
    },
    {
      "name": "Róbert Rajkó"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajt9qz5js35y5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25377",
  "fulltext_url": null,
  "id": "ajt9qz5js35y5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25377",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "openalex",
      "1602|Spectroscopy and Chemometric Analyses"
    ]
  ],
  "openalex": "W7214119539",
  "pdf_url": "https://arxiv.org/pdf/2609.25377",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajt9qz5js35y5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:49Z",
      "native_id": "2609.25377",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25377"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "Some surprising properties of essential data points visualization",
  "title_text": "Some surprising properties of essential data points visualization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:49Z",
  "url": "https://research.draconitic.com/paper/ajt9qz5js35y5",
  "venue": "arXiv",
  "versions": []
}