草图伪逆的渐近性质
Asymptotics of the Sketched Pseudoinverse
AI总结：
该研究采用随机矩阵理论分析随机草图，证明半正定矩阵正则化草图伪逆的一、二阶渐近等价性，拓展了实数场景与负正则化下的相关结果，并应用于草图投影与草图岭回归，还推广至渐近自由与正交草图矩阵。
摘要：
我们采用随机矩阵理论方法研究随机草图（random sketching），证明了半正定矩阵的正则化草图伪逆与该矩阵预解式的某一求值之间存在渐近一阶等价性。我们聚焦于实值正则化，将此前关于随机矩阵渐近等价性的结果拓展到实数场景，即便在负正则化条件下也给出了等价性的精确刻画，其中包括草图矩阵最小非零特征值的精确刻画——这一结果本身可能具有独立研究价值。随后我们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（sketch-and-project）方法分析以及草图岭回归（sketched ridge regression）。最后，我们证明这些结果可推广至渐近自由的草图矩阵，得到了正交草图矩阵的对应等价性，并将我们的结果与实践中常用的几种草图方法进行了对比。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
