Maximal Volume Matrix Cross Approximation for Image Compression and Least Squares Solution
This work provides incremental improvements to matrix cross approximation methods, benefiting researchers and practitioners in numerical linear algebra and data compression.
The authors tackled the problem of matrix cross approximation by improving the classic estimate with a better constant and introducing greedy algorithms for finding maximal volume submatrices, resulting in theoretical convergence guarantees and effective performance in applications like image compression and least squares approximation.
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.