Abstract

The article considers the problem of low-rank column and cross (\(CGR\), \(CUR\)) approximation of matrices in the Frobenius norm up to a fixed factor \(1 + \varepsilon \). It is proved that, for random matrices, in average, an estimate of the form \(1 + \varepsilon \leqslant \tfrac{{m + 1}}{{m - r + 1}}\tfrac{{n + 1}}{{n - r + 1}}\) holds, where \(m\) and \(n\) are the number of rows and columns of the cross approximation. Thus, matrices for which the maximum volume principle cannot guarantee high accuracy are quite rare. A connection of the estimates obtained with the methods for finding the submatrix of the maximum volume and the maximum projective volume is also considered. Numerical experiments show the closeness of theoretical estimates and practical results of fast cross approximation.

中文翻译：

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关于平均交叉和列低秩Maxvol近似的准确性

摘要

文章考虑了Frobenius 范数中矩阵的低秩列和交叉 ( \(CGR\) , \(CUR\) ) 逼近问题，直到固定因子\(1 + \varepsilon \)。证明，对于随机矩阵，平均而言，估计形式\(1 + \varepsilon \leqslant \tfrac{{m + 1}}{{m - r + 1}}\tfrac{{n + 1 }}{{n - r + 1}}\)成立，其中\(m\)和\(n\)是交叉近似的行数和列数。因此，最大体积原理不能保证高精度的矩阵非常罕见。还考虑了用寻找最大体积和最大投影体积的子矩阵的方法获得的估计的联系。数值实验表明了快速交叉逼近的理论估计与实际结果的接近程度。