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：尽管已有大量用于插值分解和CUR型分解的矩阵列子集选择问题（CSSP）算法，但随着输入矩阵规模增大，这些算法的应用往往变得难以实现。为解决这些问题，文献\cite{emelianenko2024adaptive}的作者开发了一个通用框架，将列分区例程与列选择算法相结合。该研究提出的四种算法中，有两种将质心Voronoi正交分解（\textsf{CVOD}）及其自适应变体（\textsf{adaptCVOD}）与离散经验插值法（\textsf{DEIM}）\cite{sorensen2016deim}进行配对。本研究扩展了这一框架，将\textsf{CVOD}型算法与任意能返回线性无关列的CSSP算法相结合。我们的研究结果不仅包含了这些配对算法所提供解的具体误差界，还明确刻画了所选列分区的质量对CSSP解的影响表达式。