  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.
