On the optimality of voronoi‐based column selection

Abstract

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, in earlier work we 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 (CVOD; Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis, 2003, 137–150) and an adaptive variant (adaptCVOD) with the Discrete Empirical Interpolation Method (DEIM; SIAM J. Sci. Computer. 38 (2016), no. 3, A1454–A1482). In this work, we extend this framework and pair the 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. In addition to examples involving matrix approximation, we test several of our partition‐based constructions on tasks commonly encountered in model order reduction (MOR).