Stochastic diagonal estimation with adaptive parameter selection

TL;DR

A lower bound on the number of random query vectors needed to satisfy a given probabilistic error bound is derived, which forms the foundation of the adaptive stochastic diagonal estimation algorithm.

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.

Figures (5)

• Figure 1: Impact of eigenvalue concentration on the accuracy of diagonal estimation based on Eq. \ref{['eq:diagonal_estimator']}
• Figure 1: The number of matrix-vector multiplications required for diagonal estimation for matrices with different eigenvalue decay patterns
• Figure 1: Comparison of different diagonal estimation methods under different spectral distributions
• Figure 2: Relative errors of diagonal estimation for matrices with different eigenvalue distribution
• Figure 2: Comparison of different methods for the number of triangles in undirected graph with respect to Wikipedia and arXiv

Theorems & Definitions (9)

• Lemma 2.1
• Proof 1
• Corollary 2.2
• Proof 2
• Lemma 2.3
• Proof 3
• Theorem 2.4
• Proof 4
• Lemma 3.1: persson2022improved, Lem. 2.2