We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results at the end of the paper demonstrate the effective performance of our approach.

我们研究基于最大体积子矩阵的经典矩阵交叉近似。我们的主要结果包括对矩阵交叉近似的经典估计的改进以及用于查找最大体积子矩阵的贪婪方法。更准确地说，我们提出了具有改进常数的经典不等式估计的新证明。此外，我们提出了一系列贪婪最大体积算法来提高矩阵交叉逼近的计算效率。所提出的算法被证明具有收敛的理论保证。最后，我们提出两个应用：图像压缩和连续函数的最小二乘近似。本文末尾的数值结果证明了我们方法的有效性能。