Sketch-and-solve is a powerful paradigm for tackling large-scale computational problems by reducing their dimensionality using sketching matrices. This paper focuses on applying sketch-and-solve algorithms to efficiently solve the overdetermined least squares problem, which is fundamental in various domains such as machine learning, signal processing, and numerical optimization. We provide a comprehensive overview of the sketch-and-solve paradigm and analyze different sketching operators, including dense and sparse variants. We introduce the Sketch-and-Apply (SAA-SAS) algorithm, which leverages randomized numerical linear algebra techniques to compute approximate solutions efficiently. Through extensive experiments on large-scale least squares problems, we demonstrate that our proposed approach significantly outperforms the traditional Least-Squares QR (LSQR) algorithm in terms of runtime while maintaining comparable accuracy. Our results highlight the potential of sketch-and-solve techniques in efficiently handling large-scale numerical linear algebra problems.

翻译：草图求解是一种通过使用草图矩阵降低问题维度来处理大规模计算问题的强大范式。本文重点研究应用草图求解算法高效解决超定最小二乘问题，该问题在机器学习、信号处理和数值优化等多个领域具有基础性意义。我们对草图求解范式进行了全面综述，并分析了包括稠密与稀疏变体在内的不同草图算子。我们提出了草图应用算法，该算法利用随机数值线性代数技术高效计算近似解。通过对大规模最小二乘问题的大量实验，我们证明所提方法在保持相当精度的同时，其运行时间显著优于传统的最小二乘QR算法。我们的研究结果凸显了草图求解技术在处理大规模数值线性代数问题方面的巨大潜力。