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

TL;DR

This paper improves the theoretical understanding and computational methods for matrix cross approximation using maximal volume submatrices, with applications in image compression and least squares approximation.

Contribution

It provides an improved estimate for matrix cross approximation, introduces greedy algorithms with convergence guarantees, and demonstrates their effectiveness in practical applications.

Findings

Enhanced estimate with better constant for matrix cross approximation

Greedy algorithms with proven convergence guarantees

Effective performance in image compression and function approximation

Abstract

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.