We take a random matrix theory approach to random sketching and show an asymptotic first-order equivalence of the regularized sketched pseudoinverse of a positive semidefinite matrix to a certain evaluation of the resolvent of the same matrix. We focus on real-valued regularization and extend previous results on an asymptotic equivalence of random matrices to the real setting, providing a precise characterization of the equivalence even under negative regularization, including a precise characterization of the smallest nonzero eigenvalue of the sketched matrix, which may be of independent interest. We then further characterize the second-order equivalence of the sketched pseudoinverse. We also apply our results to the analysis of the sketch-and-project method and to sketched ridge regression. Lastly, we prove that these results generalize to asymptotically free sketching matrices, obtaining the resulting equivalence for orthogonal sketching matrices and comparing our results to several common sketches used in practice.

翻译：我们采用随机矩阵理论的方法研究随机草图化，并证明了半正定矩阵正则化草图伪逆与该矩阵预解式在某种评估下的渐近一阶等价性。我们关注实值正则化，并将随机矩阵渐近等价性的现有结果推广至实场景，即使在负正则化条件下也给出了等价的精确刻画，包括对草图矩阵最小非零特征值的精确描述（这一结果可能具有独立价值）。随后，我们进一步刻画了草图伪逆的二阶等价性，并将成果应用于投影迭代法分析和草图岭回归。最后，我们证明这些结论可推广至渐近自由草图矩阵，获得了正交草图矩阵的相应等价性，并将结果与实践中常用的若干草图策略进行了比较。