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方法研究随机草图，证明正半定矩阵的正则化草图伪逆与该矩阵预解式的特定求值之间存在渐近一阶等价性。我们聚焦于实值正则化，并将随机矩阵渐近等价性的先前结果推广至实数域，即使在负正则化条件下也能精确刻画这种等价性——包括草图矩阵最小非零特征值的精确表征（该结果可能具有独立的研究价值）。随后进一步刻画了草图伪逆的二阶等价性。我们将这些结果应用于分析“草图-投影”方法与草图岭回归。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几种草图方法进行了对比。