PREPRINT

Asymptotics of the Sketched Pseudoinverse

Daniel LeJeune, Pratik Patil, Hamid Javadi, Richard G. Baraniuk, Ryan J. Tibshirani

Submitted on 7 November 2022

Abstract

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.

Preprint

Comment: 37 pages, 7 figures

Subjects: Mathematics - Numerical Analysis; Computer Science - Data Structures and Algorithms; Mathematics - Statistics Theory; 15B52, 46L54, 62J07

URL: http://arxiv.org/abs/2211.03751