SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 351-375, January 2021.
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 nonuniquely 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.

中文翻译：

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CUR 分解的扰动

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 1 期，第 351-375 页，2021
年1 月。 CUR 分解是通过选择低秩矩阵的某些列和行子矩阵获得的分解。我们对存在噪声的情况下这种分解会发生什么进行彻底调查。由于 CUR 分解是非唯一形成的，我们研究了几种变体，并根据噪声矩阵的幅度在包括所有 Schatten $p$-范数在内的广泛范数中给出了每个变体的扰动估计。这里给出的估计是定性的，说明了列和行的选择如何影响近似的质量，此外，我们为 CUR 近似的一些变体获得了新的最新边界。