Efficient Orthogonal Decomposition with Automatic Basis Extraction for Low-Rank Matrix Approximation

TL;DR

This paper introduces EOD-ABE, a fast and robust low-rank matrix approximation method that automatically determines the rank without prior knowledge, validated through theoretical analysis and image reconstruction applications.

Contribution

The paper presents a novel randomized algorithm for low-rank approximation that automatically extracts the basis and determines the rank, addressing a key limitation of existing methods.

Findings

Superior speed, accuracy, and robustness over existing algorithms

Effective automatic rank detection through randomized basis extraction

Successful application to image reconstruction tasks

Abstract

Low-rank matrix approximation play a ubiquitous role in various applications such as image processing, signal processing, and data analysis. Recently, random algorithms of low-rank matrix approximation have gained widespread adoption due to their speed, accuracy, and robustness, particularly in their improved implementation on modern computer architectures. Existing low-rank approximation algorithms often require prior knowledge of the rank of the matrix, which is typically unknown. To address this bottleneck, we propose a low-rank approximation algorithm termed efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) tailored for the scenario where the rank of the matrix is unknown. Notably, we introduce a randomized algorithm to automatically extract the basis that reveals the rank. The efficacy of the proposed algorithms is theoretically and numerically validated, demonstrating superior speed, accuracy, and robustness compared to existing methods. Furthermore, we apply the algorithms to image reconstruction, achieving remarkable results.