快速、高精度的随机化零空间计算用于高矩阵
Fast, High-Accuracy, Randomized Nullspace Computations for Tall Matrices
AI总结：
本文提出RLOBPCG方法，通过随机预处理与LOBPCG结合，高效计算高矩阵的最小奇异向量，实现高精度和速度提升。
摘要：
本文开发了RLOBPCG，一种高效计算大高矩阵最小奇异值对应的小数奇异三元组的方法。该算法结合了随机预处理技术中的随机预处理与LOBPCG特征求解器：一个小的草图用于构造高质量的预处理器，LOBPCG在Gram矩阵上运行以细化奇异向量。在标准子空间嵌入假设和两个最小奇异值之间有适度的奇异值间隙的前提下，我们证明RLOBPCG几何收敛到最小奇异向量。在数值实验中，RLOBPCG在具有高达10^6行的矩阵上实现了接近最优的精度，比经典LOBPCG和Lanczos方法快了高达12倍，并在其他迭代方法无法收敛时保持稳健。
英文摘要：
In this paper, we develop RLOBPCG, an efficient method for computing a small number of singular triplets corresponding to the smallest singular values of large, tall matrices. The algorithm combines randomized preconditioner from the sketch-and-precondition techniques with the LOBPCG eigensolver: a small sketch is used to construct a high-quality preconditioner, and LOBPCG is run on the Gram matrix to refine the singular vector. Under the standard subspace embedding assumption and a modest singular value gap between the two smallest singular values, we prove that RLOBPCG converges geometrically to the minimum singular vector. In numerical experiments, RLOBPCG achieves near-optimal accuracy on matrices with up to $10^6$ rows, outperforming classical LOBPCG and Lanczos methods by a speedup of up to $12\times$ and maintaining robustness when other iterative methods fail to converge.
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