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. 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.Comment: 37 pages, 7 figure