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.

本研究提出了一种高效的算法，叫做球形归一化奇异值分解 (SVD)，用于稳健的奇异值分解近似，对异常值不敏感、可扩展的计算，提供准确的奇异向量估计。该算法通过仅使用标准降秩奇异值分解算法对适当缩放的数据进行两次计算，实现了显著的计算速度，并在计算时间上明显优于竞争算法。为评估估计奇异向量及其子空间的稳健性，我们引入了矩阵型输入的新的破坏点概念，包括按行、按列和按块的破坏点。理论和实证分析表明，与标准 SVD 及其修改相比，我们的算法具有更高的破坏点。我们在高维微阵列数据集的鲁棒低秩逼近和鲁棒主成分分析等应用中，经验地验证了我们方法的有效性。总体而言，本研究提供了一种高效且稳健的 SVD 近似解决方案，克服了现有算法在异常值存在时的局限性。