Stochastic diagonal estimation with adaptive parameter selection

TL;DR

This paper introduces an adaptive stochastic diagonal estimation algorithm for large matrices, improving efficiency and stability by optimizing parameters based on eigenvalue distributions and providing theoretical bounds.

Contribution

It develops a novel adaptive stochastic diagonal estimator with parameter optimization and theoretical bounds, enhancing performance over existing methods.

Findings

The proposed estimator is more efficient than existing methods.

Numerical experiments confirm its stability across various matrices.

Theoretical analysis provides bounds on query complexity.

Abstract

In this paper, we investigate diagonal estimation for large or implicit matrices, aiming to develop a novel and efficient stochastic algorithm that incorporates adaptive parameter selection. We explore the influence of different eigenvalue distributions on diagonal estimation and analyze the necessity of introducing the projection method and adaptive parameter optimization into the stochastic diagonal estimator. Based on this analysis, we derive a lower bound on the number of random query vectors needed to satisfy a given probabilistic error bound, which forms the foundation of our adaptive stochastic diagonal estimation algorithm. Finally, numerical experiments demonstrate the effectiveness of the proposed estimator for various matrix types, showcasing its efficiency and stability compared to other existing stochastic diagonal estimation methods.