Maximal Volume Matrix Cross Approximation for Image Compression and Least Squares Solution

We study the classic cross approximation of matrices based on the maximal volume submatrices. Our main results consist of an improvement of a classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. Indeed, we present a new proof of a classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms which improve the error bound of cross approximation of a matrix in the Chebyshev norm and also improve the computational efficiency of classic maximal volume algorithm. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: one is image compression and the other is least squares approximation of continuous functions. Our numerical results in the end of the paper demonstrate the effective performances of our approach.