The CUR decomposition is a factorization of a low-rank matrix obtained by selecting certain column and row submatrices of it. We perform a thorough investigation of what happens to such decompositions in the presence of noise. Since CUR decompositions are non-uniquely formed, we investigate several variants and give perturbation estimates for each in terms of the magnitude of the noise matrix in a broad class of norms which includes all Schatten $p$--norms. The estimates given here are qualitative and illustrate how the choice of columns and rows affects the quality of the approximation, and additionally we obtain new state-of-the-art bounds for some variants of CUR approximations.
CUR Decompositions, Approximations, and Perturbations
Department of Mathematics, University of Arizona, Tucson, AZ 85719 USA（亚利桑那大学数学系，图森，亚利桑那州，85719 USA）；Department of Mathematics, Vanderbilt University, Nashville, TN 37240 USA（范德比尔特大学数学系，纳什维尔，田纳西州，37240 USA）
A Bloch decomposition based split-step pseudo spectral method for quantum dynamics with periodic potentials
Tsinghua University（清华大学）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；Wolfgang Pauli Institute Vienna（维也纳沃尔夫冈·泡利研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学学院）
