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.

翻译：低秩矩阵逼近在图像处理、信号处理和数据分析等各类应用中扮演着普遍角色。近年来，低秩矩阵逼近的随机算法因其速度、精度和鲁棒性而获得广泛采用，尤其在现代计算机架构上的改进实现中表现突出。现有低秩逼近算法通常需要预先知道矩阵的秩，而这一信息通常是未知的。为解决这一瓶颈，我们提出了一种称为"高效自动基抽取正交分解"(EOD-ABE)的低秩逼近算法，专门针对矩阵秩未知的场景。值得注意的是，我们引入了一种随机算法来自动抽取揭示矩阵秩的基。所提算法的有效性在理论和数值上均得到验证，展现出比现有方法更优的速度、精度和鲁棒性。此外，我们将算法应用于图像重建，取得了显著效果。