Stochastic Diagonal Estimation Based on Matrix Quadratic Form Oracles

TL;DR

This paper introduces a stochastic method for estimating the diagonal of an implicitly defined matrix using quadratic form oracles, with proven theoretical bounds and validated by numerical experiments.

Contribution

The paper presents a novel stochastic diagonal estimation technique leveraging matrix quadratic form oracles, with detailed complexity analysis and empirical validation.

Findings

Method effectively estimates matrix diagonals.

Theoretical bounds match empirical results.

Numerical experiments confirm efficiency across matrix types.

Abstract

We study the problem of estimating the diagonal of an implicitly given matrix $\Ab$. For such a matrix we have access to an oracle that allows us to evaluate the matrix quadratic form $ \ub^\top \Ab \ub$. Based on this query oracle, we propose a stochastic diagonal estimation method with random variable $\ub$ drawn from the standard Gaussian distribution. We provide the element-wise and norm-wise sample complexities of the proposed method. Our numerical experiments on different types and dimensions matrices demonstrate the effectiveness of our method and validate the tightness of theoretical results.