Randomized subspace approximation with "matrix sketching" is an effective approach for constructing approximate partial singular value decompositions (SVDs) of large matrices. The performance of such techniques has been extensively analyzed, and very precise estimates on the distribution of the residual errors have been derived. However, our understanding of the accuracy of the computed singular vectors (measured in terms of the canonical angles between the spaces spanned by the exact and the computed singular vectors, respectively) remains relatively limited. In this work, we present bounds and estimates for canonical angles of randomized subspace approximation that can be computed efficiently either a priori or a posterior. Under moderate oversampling in the randomized SVD, our prior probabilistic bounds are asymptotically tight and can be computed efficiently, while bringing a clear insight into the balance between oversampling and power iterations given a fixed budget on the number of matrix-vector multiplications. The numerical experiments demonstrate the empirical effectiveness of these canonical angle bounds and estimates on different matrices under various algorithmic choices for the randomized SVD.

翻译：采用“矩阵草图”的随机子空间逼近是构建大型矩阵近似部分奇异值分解（SVD）的有效方法。此类技术的性能已被广泛分析，并推导出残差分布的非常精确的估计。然而，我们对计算出的奇异向量精度的理解（分别以精确奇异向量与计算奇异向量所张成子空间之间的规范角衡量）仍然相对有限。本文针对随机子空间逼近的规范角，提出了可在先验或后验情况下高效计算的边界与估计。在随机SVD采用适度过采样时，我们的先验概率边界在渐近意义下是紧的，且可高效计算，同时能清晰地洞察在矩阵-向量乘法次数固定预算下过采样与幂迭代之间的平衡。数值实验表明，针对不同矩阵及随机SVD的多种算法选择，这些规范角边界与估计在经验上具有有效性。