发表机构
Institute for Advanced Studies in Basic Sciences (IASBS); University of Óbuda（基础科学高等研究院; 欧贝达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发现基本数据点缩减仅部分保留几何结构，未缩减子空间精确保留，缩减子空间仅非正交同构，需基变换校正才能比较。
AI 中文摘要
基本数据点（EDPs）——双线性数据矩阵 $\mathbf{D}$ 在其行空间、列空间或两者中的凸包顶点——在化学计量学中被广泛用于在名义上保留大型数据集底层几何结构的同时缩减其规模。利用模拟的三组分和两组分色谱/光谱数据集以及一个真实的源解析数据集（$\mathbf{D = C\\, A^{\mathsf{T}}}$），结合Borgen-Rajkó图、Procrustes分析和方差-协方差比较，我们证明这种保留仅是部分的：按行EDP缩减精确保留行空间几何（内多边形和外多边形），但扭曲列空间几何；按列缩减则表现出相反行为；联合按行和按列缩减则同时扭曲两者。随后，我们基于矩阵的四个基本子空间及其奇异值分解 $\mathbf{D = U\\,S\\,V^{\mathsf{T}}}$，给出一个严谨的一般性证明：未被缩减的子空间总是被精确保留（至多相差一个正交旋转），而环境维度缩减的子空间仅通过一个一般的非正交同构与原空间相关。这一区别在真实数据集上以机器精度（约 $10^{-14}$-$10^{-16}$）数值确认，且一个刻意的阴性对照证实“环境收缩”映射确实是非正交的（残差约 $\approx 1$）。这些结果表明，EDP缩减多边形相对于原始多边形的表观旋转一般不是刚性旋转，在对EDP缩减数据集与原始数据集进行任何几何或统计结论之前，需要显式的、依赖于模式的基变换校正。提供了该校正的MATLAB实现。
英文摘要
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.
