Some surprising properties of essential data points visualization
Abstract
Essential Data Points (EDPs) - the vertices of the convex hull of a bilinear data matrix 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 (D = C AT), 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 D = U S VT - 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 (∼ 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 ≈ 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.
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BibTeX
@misc{bakhshi2026some,
title = {{Some surprising properties of essential data points visualization}},
author = {Hamideh Bakhshi and Hamid Abdollahi and Róbert Rajkó},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25377},
eprint = {2609.25377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25377},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Some surprising properties of essential data points visualization
AU - Hamideh Bakhshi
AU - Hamid Abdollahi
AU - Róbert Rajkó
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25377
UR - https://arxiv.org/abs/2609.25377
AB - Essential Data Points (EDPs) - the vertices of the convex hull of a bilinear data matrix 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 (D = C A^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 D = U S V^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 (∼ 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 ≈ 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.
ER -