While there exists a rich array of matrix column subset selection problem (CSSP) algorithms for use with interpolative and CUR-type decompositions, their use can often become prohibitive as the size of the input matrix increases. In an effort to address these issues, the authors in \cite{emelianenko2024adaptive} developed a general framework that pairs a column-partitioning routine with a column-selection algorithm. Two of the four algorithms presented in that work paired the Centroidal Voronoi Orthogonal Decomposition (\textsf{CVOD}) and an adaptive variant (\textsf{adaptCVOD}) with the Discrete Empirical Interpolation Method (\textsf{DEIM}) \cite{sorensen2016deim}. In this work, we extend this framework and pair the \textsf{CVOD}-type algorithms with any CSSP algorithm that returns linearly independent columns. Our results include detailed error bounds for the solutions provided by these paired algorithms, as well as expressions that explicitly characterize how the quality of the selected column partition affects the resulting CSSP solution.

翻译：尽管存在丰富的矩阵列子集选择问题（CSSP）算法可用于插值型和CUR型分解，但随着输入矩阵规模增大，这些算法的使用往往变得不可行。为应对此问题，文献\cite{emelianenko2024adaptive}的作者开发了一个通用框架，将列分区程序与列选择算法相结合。该工作提出的四种算法中，有两种将中心沃罗诺伊正交分解（\textsf{CVOD}）及其自适应变体（\textsf{adaptCVOD}）与离散经验插值方法（\textsf{DEIM}）\cite{sorensen2016deim}配对。本研究扩展了该框架，将\textsf{CVOD}类算法与任意返回线性无关列的CSSP算法相结合。我们给出了这些配对算法所提供解的详细误差界，以及显式刻画所选列分区质量如何影响最终CSSP解的表达式。