  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.
