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.

翻译：本研究探讨了基于最大体积子矩阵的经典矩阵交叉逼近方法。主要成果包括对经典矩阵交叉逼近估计的改进以及寻找最大体积子矩阵的贪心算法。具体而言，我们提出了经典不等式估计的新证明，并改进了其中的常数项。同时，我们提出了一系列贪心式最大体积算法以提高矩阵交叉逼近的计算效率，并证明了该算法具有理论收敛保证。最后，我们展示了两个应用场景：图像压缩与连续函数的最小二乘逼近。文末的数值实验结果验证了所提方法的有效性能。