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 practical bounds and estimates for canonical angles of randomized subspace approximation that can be computed efficiently either a priori or a posteriori, without assuming prior knowledge of the true singular subspaces. 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：基于“矩阵素描”的随机子空间近似是构建大规模矩阵近似部分奇异值分解（SVD）的有效方法。此类技术的性能已得到广泛分析，并且关于残差误差分布的非常精确的估计已被推导出来。然而，我们对于计算所得奇异向量精度的理解（以精确奇异向量张成空间与计算所得奇异向量张成空间之间的典型角来度量）仍然相对有限。在本工作中，我们提出了随机子空间近似典型角的实用界与估计，这些界与估计无需真实奇异子空间的先验知识，即可先验或后验地高效计算。在随机SVD中适度过采样的条件下，我们的先验概率界是渐近紧的且可高效计算，同时清晰地揭示了在给定矩阵-向量乘法次数固定预算下，过采样与幂迭代之间的平衡关系。数值实验证明了这些典型角界与估计在不同矩阵上、针对随机SVD的各种算法选择的经验有效性。