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 propose a conjecture 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，并证明了半正定矩阵的正则化草图伪逆在渐近一阶意义上等价于该矩阵预解式的某种评估。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论推广至实数值情形，提供了即使在负正则化下也成立的精确等价刻画，包括草图矩阵最小非零特征值的精确表征（这一结果本身可能具有独立意义）。随后进一步刻画了草图伪逆的二阶等价性。我们还将所得结果应用于草图-投影方法及草图岭回归的分析。最后，我们提出一个猜想：这些结论可推广至渐近自由草图矩阵，由此获得正交草图矩阵的等价性结果，并与实践中常用的多种草图方案进行比较。