This work investigates the accuracy and numerical stability of CUR decompositions with oversampling. The CUR decomposition approximates a matrix using a subset of columns and rows of the matrix. When the number of columns and the rows are the same, the CUR decomposition can become unstable and less accurate due to the presence of the matrix inverse in the core matrix. Nevertheless, we demonstrate that the CUR decomposition can be implemented in a numerical stable manner and illustrate that oversampling, which increases either the number of columns or rows in the CUR decomposition, can enhance its accuracy and stability. Additionally, this work devises an algorithm for oversampling motivated by the theory of the CUR decomposition and the cosine-sine decomposition, whose competitiveness is illustrated through experiments.

翻译：本研究探讨了基于过采样的CUR分解的精度与数值稳定性。CUR分解通过选取矩阵的部分列与行来近似原矩阵。当选取的列数与行数相同时，由于核心矩阵中存在矩阵求逆运算，CUR分解可能变得不稳定且精度降低。然而，我们证明了CUR分解可以通过数值稳定的方式实现，并说明过采样（即增加CUR分解中选取的列数或行数）能够提升其精度与稳定性。此外，本研究基于CUR分解与余弦-正弦分解理论，设计了一种过采样算法，并通过实验验证了其竞争力。