One of the greatest success stories of randomized algorithms for linear algebra has been the development of fast, randomized algorithms for highly overdetermined linear least-squares problems. However, none of the existing algorithms is backward stable, preventing them from being deployed as drop-in replacements for existing QR-based solvers. This paper introduces sketch-and-precondition with iterative refinement (SPIR) and FOSSILS, two provably backward stable randomized least-squares solvers. SPIR and FOSSILS combine iterative refinement with a preconditioned iterative method applied to the normal equations and converge at the same rate as existing randomized least-squares solvers. This work offers the promise of incorporating randomized least-squares solvers into existing software libraries while maintaining the same level of accuracy and stability as classical solvers.

翻译：随机算法在线性代数领域最成功的应用之一，是针对高度超定线性最小二乘问题开发出的快速随机算法。然而，现有算法均不具备后向稳定性，使其无法作为现有基于QR分解求解器的直接替代方案。本文提出了带迭代精化的草图预处理方法（SPIR）与FOSSILS算法，两种可证明具备后向稳定性的随机最小二乘求解器。SPIR与FOSSILS将迭代精化与应用于法方程组的预处理迭代方法相结合，其收敛速度与现有随机最小二乘求解器相当。这项工作为将随机最小二乘求解器集成至现有软件库提供了可能，同时能保持与经典求解器同等的精度和稳定性水平。