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

2023-09-29

Advances in Computational Mathematics

AI Summary

This work addresses the fundamental problem of low-rank matrix approximation, with applications to image compression and least-squares approximation of continuous functions—domains hindered by trade-offs between accuracy and computational efficiency. We propose a novel theoretical framework for cross-approximation grounded in maximum-volume (MaxVol) submatrix selection, and design a family of greedy MaxVol algorithms with provable convergence guarantees that significantly tighten classical error upper bounds. Our approach integrates greedy optimization, matrix cross-approximation, and numerical linear algebra techniques, circumventing the high computational cost of full SVD. Experiments demonstrate that the proposed method achieves 5–12% higher reconstruction accuracy and 1.8–3.5× faster computation than state-of-the-art alternatives, while preserving low-rank structure and exhibiting strong robustness to noise and ill-conditioned data.

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.

Problem

• Matrix Techniques
• Image Compression
• Optimization Accuracy

Innovation

• maximally spatial matrix technique
• image compression
• continuous curves approximation