稳健的SVD变得简单：一种用于大规模数据分析的快速可靠算法
Robust SVD Made Easy: A fast and reliable algorithm for large-scale data analysis
AI总结：
本文提出球面归一化SVD算法，通过两次标准降秩SVD实现快速且对异常值不敏感的稳健奇异向量近似，并引入矩阵崩溃点新概念，在高维微阵列数据中验证了其优于现有算法的稳健性与效率。
摘要：
奇异值分解（SVD）是机器学习和统计分析中的重要工具。然而，它对数据矩阵中的异常值高度敏感。现有的稳健SVD算法通常为了稳健性而牺牲速度，或者在仅有少量异常值时失效。本研究提出了一种名为球面归一化SVD（Spherically Normalized SVD）的高效算法，用于稳健的SVD近似，该算法对异常值高度不敏感，在计算上具有可扩展性，并能提供奇异向量的准确近似。该算法通过仅对适当缩放的数据应用两次标准降秩SVD算法来实现显著的速度提升，在计算时间上显著优于竞争算法。为了评估近似奇异向量及其子空间对数据污染的稳健性，我们引入了矩阵值输入的崩溃点新概念，包括行向、列向和块向崩溃点。理论和实证分析表明，与标准SVD及其变体相比，我们的算法具有更高的崩溃点。我们在高维微阵列数据集的稳健低秩近似和稳健主成分分析等应用中实证验证了方法的有效性。总体而言，我们的研究提出了一种高效且稳健的SVD近似解决方案，克服了现有算法在存在异常值时的局限性。
英文摘要：
The singular value decomposition (SVD) is a crucial tool in machine learning and statistical data analysis. However, it is highly susceptible to outliers in the data matrix. Existing robust SVD algorithms often sacrifice speed for robustness or fail in the presence of only a few outliers. This study introduces an efficient algorithm, called Spherically Normalized SVD, for robust SVD approximation that is highly insensitive to outliers, computationally scalable, and provides accurate approximations of singular vectors. The proposed algorithm achieves remarkable speed by utilizing only two applications of a standard reduced-rank SVD algorithm to appropriately scaled data, significantly outperforming competing algorithms in computation times. To assess the robustness of the approximated singular vectors and their subspaces against data contamination, we introduce new notions of breakdown points for matrix-valued input, including row-wise, column-wise, and block-wise breakdown points. Theoretical and empirical analyses demonstrate that our algorithm exhibits higher breakdown points compared to standard SVD and its modifications. We empirically validate the effectiveness of our approach in applications such as robust low-rank approximation and robust principal component analysis of high-dimensional microarray datasets. Overall, our study presents a highly efficient and robust solution for SVD approximation that overcomes the limitations of existing algorithms in the presence of outliers.
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