DeepAI AI Chat
Log In Sign Up

Asymptotics of the Sketched Pseudoinverse

by   Daniel LeJeune, et al.
Carnegie Mellon University
berkeley college
Rice University

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.


How to Detect and Construct N-matrices

N-matrices are real n× n matrices all of whose principal minors are nega...

Compatible Matrices of Spearman's Rank Correlation

In this paper, we provide a negative answer to a long-standing open prob...

Gaussian Regularization of the Pseudospectrum and Davies' Conjecture

A matrix A∈C^n× n is diagonalizable if it has a basis of linearly indepe...

Contextual Equivalence for a Probabilistic Language with Continuous Random Variables and Recursion

We present a complete reasoning principle for contextual equivalence in ...

Characterization of the equivalence of robustification and regularization in linear and matrix regression

The notion of developing statistical methods in machine learning which a...

Local Equivalence Problem in Hidden Markov Model

In the hidden Markovian process, there is a possibility that two differe...