Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall-skinny matrix. We analyze a fast algorithm by Gilbert, Park and Wakin for finding the trailing right singular vectors using randomization by examining the quality of the solution using multiplicative perturbation theory. For an $m\times n$ ($m\geq n$) matrix, the algorithm runs with complexity $O(mn\log n +n^3)$ which is faster than the standard $O(mn^2)$ methods. In applications, numerical experiments show great speedups including a $30\times$ speedup for the AAA algorithm and $10\times$ speedup for the total least squares problem.

翻译：数值线性代数中的随机算法具有快速、可扩展和鲁棒的特点。本文研究了随机草图化对细长矩阵最小奇异值对应的右奇异向量的影响。我们分析了Gilbert、Park和Wakin提出的一种利用随机性计算尾随右奇异向量的快速算法，并通过乘法摄动理论考察了该算法的解质量。对于一个$m\times n$（$m\geq n$）矩阵，该算法的计算复杂度为$O(mn\log n + n^3)$，比标准的$O(mn^2)$方法更快。数值实验表明，该算法在实际应用中实现了显著的加速效果，包括在AAA算法中加速30倍，在总体最小二乘问题中加速10倍。